Test: LCM And GCD- 2 - GMAT MCQ

Step 1 & 2: Understand Question and Draw Inference

Given:

• Integers A, B > 0
  • A³ is divisible by 24
    • 24 = 2³ * 3
      • A³ is divisible by 2
        • This means, 2 is a prime factor of A
      • A³ is divisible by 3
        • This means, 3 is a prime factor of A as well
  • Thus, A is divisible by both 2 and 3
  • So, we can write: A = 2*3*k, where k is a positive integer

To find:

• Is AB² a multiple of 216?
  • 216 = 2³ *3³
• Deduced above: A is a multiple of 2*3
  • So, AB² = (2*3*k)*B²
• So, AB² will definitely be a multiple of 216 = 2³ *3³ if:
  • B² is a multiple of 2² *3²
  • That is, if B is a multiple of 2*3
• If B is not a multiple of 6, then we will need to know the values of A and B to assess if the answer to the question is YES or NO
  • Example: If A = 216 and B² = 25, then the answer will be YES
  • But if A = 36 and B² = 25, then the answer will be NO

Step 3 : Analyze Statement 1 independent

B is a multiple of 6
So, AB² will definitely be a multiple of 216
Statement 1 is sufficient to answer the question

Step 4 : Analyze Statement 2 independent

• B is divisible by 30
  • So, B is of the form 30m, where m is a positive integer
• Therefore, B is a multiple of 6
• So, AB² will definitely be a multiple of 216
• Statement 2 is sufficient to answer the question

Step 5: Analyze Both Statements Together (if needed)

Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D

Step 1 & 2: Understand Question and Draw Inference

• a is a positive integer
  •  where P₁ , P₂ are prime numbers and q, r are integers > 0
• b is a positive integer
  •   where P₃ , P₄ are prime numbers and s, t are integers > 0
• c is a positive integer
• Lowest number that has both a and b as its factors = LCM(a, b)

To Find: the value of y in c = LCM (a, b)x + y, where x and y are non-negative integers and 0 ≤ y < x

• As a and b do not have any common factor, LCM (a, b) = a* b
  • Alternatively, since a and b do not have any common prime factor, GCD(a, b) = 1
  • As a * b = LCM(a, b) * GCD(a, b), we have LCM(a, b) = 
  • c = (ab)x + y
  • Following cases can occur:
    • If c is a multiple of LCM(a, b) or ab, y = 0
    • If c is not a multiple of LCM (a, b) or ab, y ≠ 0 and y < ab

Step 3 : Analyze Statement 1 independent

1. c has the same number of factors as that of the least common multiple of a and b.

• Number of factors(c) = Number of factors (LCM(a, b)) = Number of factors (ab)
• Let  where P₅ and P₆ are prime numbers and u, v are integers > 0
  • So, (q+1)(r+1)..*(s+1)(t+1)… = (u +1) (v+1)….
• This, does not tell us for sure if c is a multiple of LCM(a, b) or not.
  • For example, 2 * 3 has the same number of factors as 5 * 7.
    However, 2 * 3 is a not a multiple of 5 * 7

Insufficient to answer.

Step 4 : Analyze Statement 2 independent

2. When c is divided by 4 times the product of a and b, the remainder is 0.

• c = 4ab
  • c is a multiple of ab, so y = 0

Sufficient to answer

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from step-4, this step is not required.
Answer: B

Step 1 & 2: Understand Question and Draw Inference

• a, b are integers > 0
  • a ≠ b

To Find: value of r in b = ax + r, where x and r are positive integers and 0 ≤ r < a

• r = 0, if a divides b completely
• 0 < r < a, if a does not divide b completely

Step 3 : Analyze Statement 1 independent

1. When b is divided by the lowest number that is divisible by both a and b, the result is an integer.

• Lowest number that is divisible both a and b = LCM(a, b)
• So, we can write:  where z is a positive integer x……(A)
• Now, we know that LCM(a, b) will have a as its factor. So, LCM(a, b) divided by a will result in an integer. So, we can write  where d is a positive integer .....(B)
• Using (A) and (B), we can write
• Comparing the above equation with b = ax + r, we see that r = 0
• Thus, the remainder will be 0
  Sufficient to answer

Step 4 : Analyze Statement 2 independent

2. a and b have the same prime factors

• You can also think of it as this way: If a smaller number is divided by a larger number the remainder will always be equal to the smaller number. For example, if 6 is divided by 18, the remainder will be equal to to 6
• So, in this case, we cannot comment on the remainder when a divides b

Insufficient to answer.

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from step-3, this step is not required.
Answer: A

Given:

• From 8 AM onwards,
  • Bell sounds after every half hour (=30 minutes)
  • Alarm rings after every 70 minutes

To Find: At which of the 5 times did the bell sound and alarm ring coincide

Approach:

1. To answer the question, we’ll first find out the time interval after which bell sound and alarm ring coincides
   1. This time interval = LCM (30, 70) minutes
2. Next, using this time interval, we’ll work out the clock times (in AM and PM) from 8 AM onwards at which the bell sound and alarm ring coincides. Then, by looking at the answer choices, we’ll be able to determine the correct answer choice

Working out:

• Finding the time-interval after which the bell and alarm coincide
  • LCM(30, 70) = LCM(2*3*5, 2*5*7) = 2*3*5*7 = 210 minutes = 3 hours 30 minutes
  • So, the bell and alarm coincide after every 3 hour 30 minutes
• Finding the clock times at which the bell and alarm coincide
  • Counting 3 hour 30 minute intervals from 8 AM onwards, these times will be:
    • 11:30 AM, 3 PM, 6:30 PM, 10 PM

Looking at the answer choices, we see that the correct answer is Option C

Given:

• The number of red balls, say r = 126
• The number of blue balls = m
• m is completely divisible by 18
  • m is also divisible by 6 other numbers greater than 1
  • So, total factors of m = {1, 18, 6 other factors greater than 1}
  • So, total number of factors of m = 2 + 6 = 8

To Find: When 126 and m balls are distributed equally among cardboard boxes, what is the minimum number of (red + blue) balls per box?

Approach: 

1. The minimum number of balls will happen when the number of cardboard boxes used is the maximum
2. So, to answer the question, we first need to find the maximum number of cardboard boxes that can be used 

• The 126 red balls can be distributed equally among a number of boxes only if the number of boxes is a factor of 126
  • For example, can you split 126 balls equally between 2 boxes? Yes.
  • Because 2 is a factor of 126
  • Can you split 126 balls equally between 10 boxes? No. Because 10 is not a factor of 126.
• Similarly, m blue balls can be distributed equally among a number of boxes  only if the number of boxes is a factor of m
• Since both red and blue balls need to be distributed equally, the number of boxes must be a common factor of m and 126
  So, maximum number of boxes = GCD(m, 126)

2. To find GCD(m, 26), we need to know the value of m.

• We’ll use the clues given about m:
  • m is of the form 18k
  • The total number of factors of m is 8

Working out:

• Finding the value of m
  • We are given that m is of the form 18k
    • 18 = 2*3²
    • So, m = 2*3² *k
    • So, m has a minimum of 2 prime factors

• Also, m has 8 factors
  • Either 8 = (0+1)(7+1)
    • That is, m is of the form P
    • Rejected, because we’ve noted above that m definitely has at least 2 prime factors
• Or 8 = (1+1)(3+1)
  • That is, m is of the form P₁*P₂³
  • We already know that m definitely does contain 2 and 3
  • So, the only possible value of m is 2*3³

Thus, m = 2*3

• Finding GCD(m, 126)
  • 126 = 2*3²*7
  • As inferred above, m = 2*3³
  • So, GCD(m, 126) = 2*3² = 18
     
• Getting to the final answer
  • The maximum number of cardboard boxes that can be used = 18
    In this case, the number of:

So, the total number of balls per box = 7 +3 = 10
Looking at the answer choices, we see that the correct answer is Option B

Given:

• Let the 2 positive integers be A and B
• A*B = 540

To Find: Which of the 3 pairs of numbers can be LCM(A,B) and GCD(A,B) respectively

Approach:

1. To answer the question, we’ll evaluate the constraints on LCM(A,B) and GCD(A,B):
   1. Constraint 1: LCM(A,B) * GCD(A,B) = A*B
   2. Constraint 2: The LCM(A,B) contains the highest power EACH prime factor of A and B.
      • This means, every prime factor that occurs in either one or both of A and B is represented in the LCM(A,B)
        1. So, every prime factor that occurs in the product of A and B will also occur in LCM(A,B)
2. Next, we’ll find which of the 3 pairs satisfy both the above constraints

Working out:

• Evaluating the 2 constraints
  • Constraint 1: LCM(A,B) * GCD(A,B) = 540
  • Constraint 2:
    • 540 = 2² *3³ *5
      • The prime factors of 540 are 2, 3 and 5
    • So, the prime factors of LCM(A,B) are 2, 3 and 5
• Checking the 3 pairs
  • 108 and 5
    • The product of 108 and 5 is 540. So, the first Constraint is satisfied
    • 108 = 2 *3
      • The prime factors of 108 are not 2, 3 and 5. So, Constraint 2 is not satisfied
      • So, this pair is rejected

• 90 and 6
  • The product of 90 and 6 is 540. So, the first Constraint is satisfied
  • 90 = 2*3 *5
    • The prime factors of 90 are 2, 3 and 5. So, Constraint 2 is also satisfied
  • So, this pair is possible

• 27 and 20
• The product of 27 and 20 is 540. So, the first Constraint is satisfied
• 27 = 3³
  • The prime factors of 27 are not 2, 3 and 5. So, Constraint 2 is not satisfied
• So, this pair is rejected

Looking at the answer choices, we see that the correct answer is Option B

Given:

2 numbers : 

To Find: GCD  

Approach:

1. order to find the GCD of the 2 numbers, we first need to find the expressions for each number
2. So, using the given definition of the factorial (!) function, we will find the expression of each number.

Working out:

Looking at the answer choices, we see that the correct answer is Option C

Step 1 & 2: Understand Question and Draw Inference

• and d = , where P , P , P and P are prime number and q, r, s, and t are integers > 0

To Find : Number of prime factors of b*d

• b* d = 
• If P , P , P and P are distinct prime numbers, then b*d will have 4 prime factors, else b *d will have less than 4 prime factors
  • If b and d have 1 shared prime factor, then b *d will have 3 prime factors
  • If b and d have 2 shared prime factors, then b*d will have 2 prime factors

Step 3 : Analyze Statement 1 independent

1. The least common denominator of  a/b and c/d is half the product of b and d
    

• Least common denominator of a fraction means the LCM of the denominators.
• LCM(b, d) = 
  • 2* LCM(b, d) = bd ...(A)
• Now, from the property of GCD and LCM, we know that GCD(b, d) *LCM(b, d) = b*d . . . (B)
  • So, using equations (A) and (B) together, we can write:
    • GCD(b, d) * LCM(b, d) = 2*LCM(b, d)
    • GCD(b, d) = 2

• So, 2 divides both b and d. Hence, 2 has to be the only common prime factor of both b and d. Had there been any other common prime factor, it would have come in the GCD(b, d)
  • Let’s assume P = P = 2
• As b and d have 1 common prime factors, therefore the number of prime factors of b*d = 2, P and P , i.e. a total of 3 prime factors.

Sufficient to answer.

Step 4 : Analyze Statement 2 independent

2. The highest integer that divides both b and d completely is 2

• That is, GCD(b, d) = 2
• So, 2 divides both b and d. Hence, 2 has to be the only common prime factor of both b and d. Had there been any other common prime factor, it would have come in the GCD(b, d)
  • Let’s assume P = P = 2
• As b and d have 1 common prime factor, therefore the number of prime factors of b*d = 2, P and P , i.e. a total of 3 prime factors.

Sufficient to answer

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D

Step 1 & 2: Understand Question and Draw Inference

• n! = 1*2*3……*n
• x, y are distinct integers > o
• y > x

To Find: Values of x and y

Step 3 : Analyze Statement 1 independent

1. The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.

• x! = 1*2*3…..*x
• y! = 1*2*3….x*(x+1)…….* y ( as y > x)
  • So, we can write: y! = x! * a, where a is a positive integer
• As y! is a multiple of x!, LCM( x!, y!) = y!
• Similarly, as x! is a factor of y!, GCD(x!, y!) = x!

So,

Hence, 20 needs to be expressed as a product of 1 or more consecutive positive
integers. Following cases are possible:

• As 20 = 2² * 5, following cases are possible:
• 20 = 4 * 5
  • So, (x+1) * y = 20. Therefore, x + 1 = 4, i.e. x = 3 and y = 5
• The other possible case is when (x+1) = y = 20, i.e. y = 20 and x = 19. So,

As we do not have unique values of x and y, the statement is insufficient to answer

Step 4 : Analyze Statement 2 independent

2. The lowest common multiple and the highest common factor of y! and x!
have 3 and 2 prime factors respectively.

• we know that LCM(x!, y!) = y! = 1*2*3*4*5…y (we deduced this in the analysis of statement-1)
  • As y! has 3 prime factors only, they have to be the smallest 3 prime factors, i.e. {2, 3 , 5}. So, 5! ≤ y! < 7! as 7! will have 4 prime factors {2, 3, 5, 7}
  • 5 ≤ y < 7. So, y = 5 or 6
• We also know that GCD(x!, y!) = x! = 1*2*3… x (we deduced this in the analysis of statement-1)
  • As x! has 2 prime factors only, they have to be the smallest 2 prime factors , i.e. {2, 3}. So, 3!≤ x! < 5!, as 5! will have 3 prime factors {2, 3, 5}
  • 3≤ x < 5 So, x = {3, 4}

As we do not have unique values of x and y, insufficient to answer

Step 5: Analyze Both Statements Together (if needed)

1. From statement-1, we inferred that y = {20, 5} and x = {19, 3}
2. From statement-2, we inferred that y = { 5, 6} and x = { 3,4 }
Combining both the statements, we have y = 5 and x = 3.
Sufficient to answer.
Answer: C

Given:

• Justin started from point A and Joe started from point B traveling eastwards
  • Point B was 8 kilometres eastwards of point A

• Justin stopped for break after every 12 kilometres
• Joe stopped for break after every 10 kilometres

To Find: Number of common points within a distance of 230 kilometres from point A where Justin and Joe both stopped for a break?

Approach:

1. Had both Justin and Joe started from the same point, they would have taken a break at the same point at an interval of every LCM (10, 12) kilometres.
2. However as Joe started from a point, that was 8 kilometres east of Justing, our first task would be to find the first common point at which Justin and Joe stopped.
3. From the first common point, Justin and Joe would break at the common points at an interval of every LCM (10, 12) kilometres.

Working out:

1. Justin would take a break at 12, 24, 36, 48, 60..kilometres from point A
2. Joe would take a break at 10, 20, 30, 40, 50, 60…. Kilometres from point B
3. However, as point B is 8 kilometres east of point A, Joe would take a break at (10+8, 20+8, 30+8, 40+8…) kilometres from point A
4. Comparing the points at which Justin and Joe take a break, we can see that the first common point at which both of them took a break is 48 kilometres from point A.
5. Now, LCM(10, 12) = 60
6. So, Justin and Joe would take a break at points (48+60), (48+120), (48+180) kilometres from point A. All the other points would be outside the range of 230 kilometres from point A
7. So, Justin and Joe took a break at 48, 108, 168, 228 kilometres from point A, i.e. a total of 4 points.

Answer : D

Step 1 & 2: Understand Question and Draw Inference

Given: Integer z > 0
To find: Number of factors of z

• Let z = P *P *P * . . . , where P , P , P etc. are prime numbers and a, b, c . . . are non-negative integers
• So, number of factors of z = (a+1)*(b+1)*(c+1)* . . .
• So, in order to be able to apply the formula for number of factors, we need to know the prime-factorized form of z

Step 3 : Analyze Statement 1 independent

Statement 1 says that ‘z/5 and z/7 and are integers and the greatest integer that divides them both is 8’

• Since both z/5 and z/7  and are integers, this means, z contains 5 and 7 for sure

• Upon comparing the form 5^a-1*7^b-1 * P₃^c* . . . with 2³ (which can be rewritten as 2³ *5⁰ *7⁰ for easier comparison) , we see that:
  • P₃ = 2 and c = 3
  • a – 1 = 0; So, a = 1
  • b – 1 = 0; So, b = 1
  • And z has no other prime factor (else it would have appeared in the  GCD of  z/5 and z/7
  • So, z = 2³*5*7
  • Since we now know the prime-factorized form of z, we can answer the question
  • Statement 1 is sufficient

Step 4 : Analyze Statement 2 independent

Statement 2 says that ‘The smallest integer that is divisible by both z and 14 is 280’

• So, LCM(z, 14) = 280
  • 280 = 2³*5*7
• 14 = 2*7
• The power of 2 in 14 is only 1. Therefore, the 2 term in LCM(z, 14) must come from z.
• 14 does not contain any 5. So, the 5 in LCM(z, 14) must come from z
• 14 contains 7¹, same as the LCM(z, 14) does. So, the power of 7 in z may be 0 or 1
• Therefore, either z = 2³*5*7⁰ or z = 2³*5*7¹
• Correspondingly, the number of factors of z = Either (3+1)*(1+1)*(0+1) or (3+1)*(1+1)*(1+1)
• Since Statement 2 leads us to two possible values for the number of factors of z, it is not sufficient to get a unique answer

Step 5: Analyze Both Statements Together (if needed)

Since we’ve already arrived at a unique answer in Step 3, this step is not
required
Answer: Option A

Given:

• a, b are distinct integers > 0 such that b > a

• Please note that a and b may or may not share any or all common prime factors. So, there may be a case where a and b have the same set of prime factors but with different powers, for example (a, b) = (6, 36) OR
• a and b share some prime factors with same or different powers, for example (a, b) = ( 6, 18) .

To Find:

Approach:

c. Hence, the maximum value of the ratio   will be when GCD (a,b)² is minimum, i.e. when GCD(a, b) is minimum

1. Please note that we are not considering the possible values of the product ab, because for a number ab there can be more than 1 way to express it as a product of two numbers. For example, 18 = (2 * 9), (3* 6), (18 * 1). The minimum GCD is possible for the set (2, 9) and (18, 1), which is 1.
2. So, we will focus our attention on maximizing and minimizing the value of GCD(a, b)

d. Also, the ratio    will have its minimum value when  GCD (a,b)² is maximum, i.e. when GCD(a, b) is maximum

Working out:

1. Finding maximum value of the ratio 

• Minimum value of GCD(a, b) is possible when a and b do not have any common prime factors.

2. Minimum value of the ratio  

• Maximum value of GCD(a, b) is possible when a and b have the same prime factors.?

Answer C

Step 1 & 2: Understand Question and Draw Inference

Given: A list with 4 distinct positive integers

To find: w – x

Step 3 : Analyze Statement 1 independent

• So, from (I) x + w = 32
  • w=32-x
• So, w-x=32-2x

Since we don’t know the value of x yet, we cannot find a unique value of 32 – 2x.
Statement 1 is not sufficient.

Step 4 : Analyze Statement 2 independent

2. The greatest number that divides each of the integers in the list completely is 4.

• GCD of x, y, z and w is 4
  • So, x, y, z and w are all multiples of 4
  • The sum of these multiples of 4 is 60

• Thus we see that multiple values of w – x are possible

So, Statement 2 alone is not sufficient to answer the question

Step 5: Analyze Both Statements Together (if needed)

• From Statement 1: y + z = 28 and x + w = 32
• From Statement 2: GCD(x, y, z, w) = 4
• By combining both statements, two ordered sets of possible values of x, y, z and w are obtained:
  • {4 8, 20, 28}
    • w-x=28-4=24
  • {8, 12, 16, 24}
    • w-x=24-8=16

(Note: x cannot be 12 because then y = 16 and 28 – 16 becomes less than 16
which cannot be, since the order of the integers is x < y < z < w)
Since we get two values of the range, combining both statements is not
sufficient either to answer the question.
Answer: Option E

Given:

• Integer x > 0
  • x is a multiple of 14
  • x is a multiple of 12

To find: Which of the 3 statements must be true?

Approach:

1. A statement will be a must be true statement if it holds true for all values of x (no exceptions)
2. In order to evaluate the 3 statements, we will first use the given information to work out an expression for x. Then, we’ll evaluate the 3 statements one by one.

Working out:

Finding the expression for x

• Since x is a multiple of both 14 and 12, x will be a multiple of the LCM(14,12)
  • 14 = 2*7
  • 12 = 2²*3
  • So, LCM(14, 12) = 2² *3*7
• Therefore, x will be a multiple of 2³ *3*7
  • So, we can write: x = 2² *3*7*k, where k is a positive integer
• Evaluating Statement I
  • As per Statement I, x is divisible by 8
    • 8 = 2³
• We’ve worked out above that x = 2 *3*7*k
  • If k = 1, then x is NOT divisible by 8
  • If k = 2, then x = 2 *3*7, and therefore is divisible by 8
  • If k = 3, then x is NOT divisible by 8
  • And so on . . .
  • We see that Statement I will hold only for even values of k, not for all values of k.
  • Therefore, Statement I does not hold true for all possible values of x
  • So, Statement I is NOT a must be true statement
• Evaluating Statement II
• As per Statement II, x is divisible by 28
  • 28 = 2² *7
• We’ve worked out above that x = 2² *3*7*k
• So, it is clear that x is indeed divisible by 2² *7
• So, Statement II is a must be true statement.

• Evaluating Statement III
• As per Statement III, the greatest common divisor of x and 9 is 3
  • 9 = 3²
  • x = 2² *3*7*k
  • If k = 1, then GCD(x, 9) = 3
  • If k = 2, then GCD (x, 9) = 3
  • If k = 3, then x = 2² *3² *7 and GCD(x,9) = 3²
  • And so on...
• Thus, we see that the value of the GCD(x,9) depends on the value of k
• So, Statement III does not hold true for all values of x
• So, it is NOT a must be true statement

• Getting to the answer
  • Through our analysis, we’ve seen that only Statement II is a must be true statement
  • Looking at the answer choices, we see that the correct answer is Option B

Given:

• Positive integers M and N
• GCD (M, N) = 1
  • As GCD(M, N) = 1, that means M and N do not share any common prime factors

To Find: Which of the 3 statements are true for ALL values of M and N? (A must be true statement is true for ALL values of M and N. If even one value of M and N doesn’t satisfy a statement, then it is not a must be true statement)

Approach: 

1. ,We will evaluate each of the given 3 statements to determine which is a must be true statement

Working out:

• Evaluating Statement I
• The least common multiple of M and N has four factors

• 
• Since GCD(M,N) = 1, LCM(M,N) = M*N
• If the product M*N is of the form P *P or of the form P , where P₁ and P₂ are prime numbers, then Statement I will hold true
  • For example, when M = 2 and N = 5
  • Or when M = 1 and N = 2
  • (Note: a case like M = 2 and N = 2 is not possible because then GCD(M,N) will not be 1)
• But for other values of M and N, Statement I will not hold true.
  • For example, when M = 3 and N = 2
• So, Statement I is not a must be true statement

• Evaluating Statement II
• M and N have opposite even-odd nature 
• ?The fact that GCD(M, N) = 1 indicates that they are not both even (because if M and N were both even, then they would both be divisible by 2. So, their GCD would have been 2 or another even number then)
• However, it does not necessarily mean that M and N have opposite even-odd nature
  • M and N may have even-odd nature. Example, M = 8 and N = 9
  • Or, M and N may be both odd. Example, M = 19 and N = 21
  • So, Statement II is not a must be true statement

• Evaluating Statement III
• M = N + 1
  • For any pair of consecutive numbers, the GCD is equal to 1
  • But the vice-versa is not true. So, if GCD(M,N) = 1, this does not necessarily mean that M and N are consecutive integers. For example, M could be 19 and N could be 21
  • Thus, Statement III is not a must be true statement

Looking at the answer choices, we see that the correct answer is Option E 

Step 1 & 2: Understand Question and Draw Inference

• x is a prime number and y is an integer > 0
• x, y have two common factors
  • All integers have one common factor i.e. 1
  • Since x is prime, the other number that divides both x and y has to be x.
  • As x is a factor of y, we can write y = ax, where a is a positive integer

• Therefore, to answer the question, we need to find the ratio y/x

Step 3 : Analyze Statement 1 independent

1.  y is divisible by 4 positive integers and has 2 prime factors, one of which is 5.

•  y = 5 * P , where P is a prime number. Two cases are possible

As we do not have a unique value for  y/x  the statement is insufficient to answer

Step 4 : Analyze Statement 2 independent

2. y is equal to 3 times the highest number that divides both x and y.

Sufficient to answer

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from step-4, this step is not required.
Answer: B

Step 1 & 2: Understand Question and Draw Inference

• a and b are distinct integers > 0

To Find: Highest number that leaves no remainder when it divides a and b?
The question is asking us to find the highest common multiple of a and b, i.e. GCD(a,b)?

Step 3 : Analyze Statement 1 independent

1. a and b can be written as 8x and 8y respectively, where x and y are distinct integers with no common prime factors.

• Since a and b do not have any other common factor apart from 2³, GCD(a, b) = 8

Sufficient to answer.

Step 4 : Analyze Statement 2 independent

2. a is equal to 24 and both a and b are divisible by the same perfect cubes.

• a is divisible by 2 perfect cubes, 1 and 8
• As b is also divisible by the same perfect cube as a, we can write
• So, a and b have 2 as one of their common prime factor, however we do not know if 3, which is a factor of a is also one of the factors of b. If 3 is one of the prime factors of b, GCD(a, b) = 24
• If 3 is not one of the prime factors of b, GCD(a, b) = 8 Insufficient to answer.

Step 5: Analyze Both Statements Together (if needed)

As we have a unique answer from step-3, this step is not required.
Answer: A

Step 1 & 2: Understand Question and Draw Inference

Given:

• Prime Number P
• Integer Q > 0

To find:

• GCD(P, Q)
  • If prime number P is a prime factor of Q, then GCD(P,Q) =P
    • Example: say P = 5 and Q = 2 *5 . Then GCD (P,Q) = 5
  • If prime number P is not a prime factor of Q, then GCD(P,Q) = 1
    • Example: say P = 5 and Q = 3*7. Then GCD(P,Q) = 1
• So, in order to answer the question, we need to know:
  • If P is a prime factor of Q
  • If yes, then we also need to know the value of P
  • If no, then the answer is 1

Step 3 : Analyze Statement 1 independent

• LCM(P,Q) = 12
  • 12 = 2² *3
  • Thus, the prime factors of LCM are 2 and 3 only
    • That is, the prime factors contained in P and Q combined are
    • 2 and 3 only
• Since P is a prime number, possible values of P = {2, 3}

• Case 1: P = 2
  • LCM(2, Q) = 12
  • The only value of Q for which this equation is true: Q = 12
  • P is a factor of Q in this case
  • So, GCD (P,Q) = P = 2
• Case 2: P = 3
  • LCM(3, Q) = 12
  • The values of Q for which this equation is true:
    • Q = 12
      • P is a factor of Q in this case
      • So, GCD (P,Q) = P = 3
    • Q = 4
      • P is not a factor of Q in this case
      • So, GCD (P,Q) = 1
• Thus, we get three possible values of the GCD(P,Q) from Statement 1: {1, 2, 3}

Statement 1 alone is not sufficient to find a unique answer to the question

Step 4 : Analyze Statement 2 independent

• The product of P and Q has 6 factors
• From the formula for total number of factors, the number of factors can be 6 in 2 cases:
  • Case 1: 6 = 1*6
    • 6 = (0+1)(5+1)
    • Thus, in this case, the product PQ has only one prime factor and the power of that prime factor is 5
      • But P is given to be a prime number
      • This means, the only prime factor of PQ is P
    • So, PQ = P⁵
      • Q = P⁴
    • Thus, P is a prime factor of Q
    • So, GCD(P,Q) = P
    • But we do not know the value of P
    • So, we cannot find the GCD(P,Q) in Case 1
  • Case 2: 6 = 2*3
  • 6 = (1+1)(2+1)
  • Thus, in this case, the product PQ has two prime factors - the power of one prime factor is 1 and the power of the other prime factor is 2
    • We know that P is a prime number
      • So, P is one of the 2 prime factors of PQ. Let the other prime factor be P
    • We do not know if 2 is the power of P or of P

• Subcase 1: PQ = P*P
  • Q = P₂²
  • So, P is not a prime factor of Q in this subcase
  • So, GCD (P, Q) = 1
• Subcase 2: PQ = P * P
  • So, Q = P*P₂
  • So, P is a prime factor of Q in this subcase
  • Therefore, GCD (P, Q) = P
  • But we do not know the value of P
  • So, we cannot find the GCD(P,Q) in Subcase 2

Thus, even after analyzing Statement 2, both the possible values of GCD(P, Q) remain: Either GCD(P,Q) is equal to 1 or it is equal to P (and we don’t know the value of P)
So, Statement 2 is not sufficient to answer the question

Step 5: Analyze Both Statements Together (if needed)

From Statement 1: 3 possible values of P and Q were obtained:
i. P = 2 and Q = 12

• a. GCD (P,Q) =
• b. PQ = 2*12 = 24 = 2 *3
• c. Number of factors of PQ = (3+1)(1+1) = 4*2 = 8

ii. P = 3 and Q = 12

• a. GCD (P,Q) = 3
• b. PQ = 3*12 = 36 = 2 *3
• c. Number of factors of PQ = (2+1)(2+1) = 3*3 = 9

iii. P = 3 and Q = 4

• a. GCD (P,Q) = 1
• b. PQ = 3*4 = 12 = 3*2
• c. Number of factors of PQ = (1+1)(2+1) = 2*3 = 6

• From Statement 2: The product of P and Q has 6 factors
• Out of the 3 cases obtained from Statement 1, the only case in which Statement 2 is satisfied is Case iii: P = 3 and Q = 4
• In this case, GCD (P, Q) = 1

Thus, both the statements together are sufficient to find a unique value of the GCD.
Answer: Option C

Given:

• Let the total number of students be n
• When n is divided by 10, the remainder is 5
  • So, n is of the form 10k + 5, where k is a non-negative integer
    • k is a non-negative integer because for negative values of k, n will be negative as well, which is not allowed since the number of students cannot be negative.
• When n is divided by 21, the remainder is 0
  • So, n is also of the form 21j, where j is a positive integer

To Find: Can number of rows be 15, 21 or 30?

Approach: 

1. Number of rows =n/21
   1. So, to answer the question about the possible values for number of rows, we need to know the possible values of n.
2. One way to answer this question is to calculate the value of n corresponding to each of the 3 values for number of rows (15, 21 or 30) and then check these values of n for the constraints on value of n (n is of the form 10k + 5 and 21j)
3. Another way to answer is to first deduce a general expression for n that satisfies both the given constraints on n. Then, deduce a corresponding general expression for ‘number of rows’. This way, we’ll be able to see which of the 3 values satisfies the general expression for ‘number of rows’. We will illustrate this second approach in the working out section.

Working out:

• Finding a general expression for n
  • The 2 expressions given about n are:
    • n = 10k + 5 = 5(2k + 1)
      • This means, n is the product of 5 and a positive odd integer
    • And, n = 21j = 3*7j
      • This means, 3 and 7 are definitely the prime factors of n
  • So, combining both expressions, we can write:
    • n = 3*5*7h, where h is a positive odd integer
  • So, from the general expression of n, we learn that n is odd and definitely divisible by 3, 5 and 7
• Finding a general expression for ‘number of rows’
  • Number of rows =  where h is a positive odd integer
  • Thus, the ‘number of rows’ is an odd multiple of 5

• Evaluating the 3 values
  • Looking at the 3 values, we see that only value I satisfies the inference that ‘number of rows’ is an odd multiple of 5

Looking at the answer choices, we see that the correct answer is Option A

Given:

To Find: Number of possible values of n

Approach:

• To find the number of possible values of n, we will make use of the constraints on n
  • Constraint 1: n is completely divisible by 12 and 8
    • i. So, our first step will be to find the LCM of 12 and 8
    • ii. n will be a multiple of this LCM
    • iii. So, we will list the first few multiples of the LCM. These will all be the possible values of n according to Constraint 1.
  • Constraint 2: 25 < n < 64
    • _​​From the list of possible values of n obtained from Constraint 1, we will see how many values lie between 25 and 64, exclusive. That will give us the answer.

Working out:

• Applying Constraint 2:
  • As noted in the Approach, 25 < n < 64
  • From Constraint 1, we know that the possible values of n are {24, 48, 72, 96, 120 . . .}
  • Out of these, only 1 value, 48, lie between 25 and 64
• Getting to the answer:
• Thus, we see that only 1 value of n satisfies both Constraints 1 and 2

Therefore, the correct answer is Option B – 1.