Test: LCM And GCD- 1 - GMAT MCQ

Given:

• m is a positive integer and n is an even prime number.

Find:

• The GCD of 20m and 6n.

Working Out:

• It is given that n is an even prime number, and we know there is only one even prime number
• Hence, n = 2 and 6n = 12.

• Therefore, we can rewrite the question statement as -
  • Find the GCD of 20m and 12.

• Per our conceptual understanding, to find the GCD, we first need to perform the prime factorization of the two numbers.
  • Therefore, 20m can be written as
    • 2² x 5¹ x m
  • 6n = 12 can be written as
    • 12 = 2² x 3¹
• Next, we need to consider the lowest power of the common prime factors between 20m and 6n to find the GCD

• We can clearly notice that 20m and 12 have a minimum factor of 2² common between them.
• Since we don’t know the value of m we cannot say for sure if there is any common factor other than 2²
• Thus the only conclusion that we can make is that the GCD of 20m and 6n is a multiple of 2² = 4.

Since we need to find, out of the given options which of them “could” be the value of the GCD, we simply have to look for an option which is a multiple of 4.

There is only one of them, which is option B and hence we can conclude that the GCD of 20m and 6n could be 4.

Correct Answer: Option B

Step 1: Question statement and Inferences

We are given a positive integer p. The greatest common divisor of:

1. 25 and p is 5
2. 56 and p is 7

So, here we have to find the value of p. 

Step 2: Finding required values

Now, let’s list the prime factors of 25 in exponential form:

25 = 5²  

Since the greatest common divisor of 25 and p is 5, p can be represented as follows:

p = 5¹ * k₁         (where k₁ is a positive integer)

Next, listing the prime factors of 56 in exponential form, we get:

56 = 2³ * 7    

Since the greatest common divisor of 56 and p is 7, then p can be represented as follows: 

p = 7¹ * k₂       (where k₂ is a positive integer)

So, we know that the number p is a multiple of 5 and 7. 

Thus, we can write it as:

p = 5¹ * 7¹ * k₃ = 35*k₃           (where k₃ is a positive integer)

Step 3: Calculating the final answer

Now, let’s check the options to see if there is any option that is in the format 35*k₃. As we can see, option D satisfies this condition if we put k₃ = 1. So, option D is the correct answer. 

Answer: Option (D)

Step 1: Question statement and Inferences

Given:   A, B are positive integers

                LCM (A,B) = 72

      To find: The remainder when the product AB is divided by 8

--> We need to find the value of AB

Step 2: Finding required values

We know that product of two numbers = LCM * GCD

Therefore, AB = LCM * GCD = 72 * GCD

 Step 3: Calculating the final answer

Now AB = 72 * GCD

As 72 is divisible by 8, (72 * GCD) will also be divisible by 8.

So, AB is divisible by 8

Hence, Remainder = 0

 Answer: Option (A)

Step 1: Question statement and Inferences

We are required to find the value of Z

Z can be expressed in form of P₁^a * P₂^b * …… P_nⁿ , where P₁, P₂ . . . are prime numbers and a, b . . . are non-negative integers

Hence, we need to find all the prime factors and their powers in Z

Step 2: Finding required values

Given: LCM of Z and 24 is 72

We reverse the steps of finding LCM and find the highest power of all the prime factors in 24 or Z

LCM (Z, 24) = 72 = 3² * 2³

-->  The highest power of 3 in the given numbers is 2

The highest power of 2 in the given numbers is 3

As, 24 = 3 * 2³

-->  24 does not contain 3²

-->  Z must contain 3²

Z may or may not contain 2, 2² or 2³

Given: GCD of Z and 54 is 9

We reverse the steps of finding GCD and find the lowest power of all the prime factors in 54 and Z

GCD (Z, 54) = 9 = 3²

As 54 = 2 * 3³  => Hence the lowest power of 3 in Z is 2

=> Z must contain 3² = 9 and it should not contain any power of 2(else it should have appeared in the GCD of Z and 54).

Step 3: Calculating the final answer

As z contains only 3² and no other prime factors, z = 9.

Answer: Option (C)

Steps 1 & 2: Understand Question and Draw Inferences

We are given that for two positive integers x and y:  

1. x > y   
2. Least common multiple of x and y = 36    
3. Greatest common divisor of x and y = 6  

Now, we know that for two numbers:

Least common multiple * greatest common divisor = Product of the numbers

So, x * y = 36 * 6 = 216   ………………. (1)

However, this information can’t directly help us to find the value of x and y. So let’s move on to the analysis of statement 1 and 2. 

Step 3: Analyze Statement 1

Statement 1 says: x = 18 

By plugging the value of x in equation (1), we get:

18 * y = 216

y =12

So, x - y =6

Hence, statement I is sufficient to answer the question: What is the value of x -y?  

 Step 4: Analyze Statement 2

Statement 2 says:  x + y = 30    …………. (2)

 We know that:

(x – y)² = x² + y² – 2xy

By adding and subtracting 2xy in the right hand side, we get:

(x – y)² = x² + y² – 2xy + 2xy -2xy 

(x – y)² = x² + y² + 2xy -4xy 

(x – y)² = (x + y)² – 4xy

Plugging in the values from equation (1) and (2):

(x – y)² = (30)² – 4*216  

               = 900 – 864

               = 36 

So, x – y = 6

Hence, statement II alone is sufficient to answer the question: What is the value of x -y?  

Step 5: Analyze Both Statements Together (if needed)

Since statement I and II alone are sufficient to answer the question, we don’t need to perform this step.

Answer: Option (D)  

Step 1: Question statement and Inferences

We are given that p and q are positive integers. And we have to find that option among the five values given that cannot be the greatest common divisor of 48p and 18q. 

Step 2: Finding required values

We know that in order to find the GCD of two numbers, we should first find the prime factors of the numbers and then list those prime factors in exponential form.

We do not know the value of p and q, but at least we can factorize 48 and 18. Therefore, we can write the numbers 48p and 18q as:

                                                                     48p=3×24×p

                                                                    18q=2×32×q

Step 3: Calculating the final answer

We know that in order to find the GCD of two numbers, we pick up the LOWEST power of each prime factor that occurs in the numbers.

 The lowest power of 2 that occurs in 48p and 18q is 2¹

And, the lowest power of 3 is 3¹

Thus the GCD of the two numbers 48p and 18q will definitely contain 21×31=6

 Thus, the GCD of 48p and 18q will be divisible by 6.

Out of the given options, only Option E contains a number that is not divisible by 6. 

Answer: Option (E)

Steps 1 & 2: Understand Question and Draw Inferences

We are given two positive integers x and y. We have to find the greatest common divisor of these integers.

Since there is no other information provided in the question, let’s move on to the analysis of statements 1 and 2.    

Step 3: Analyze Statement 1

Statement 1 says: x = y + 6  

This statement tells us that the difference between the values of x and y is 6. Since it doesn’t provide any information about the values of x and y, we can’t find the greatest common divisor of x and y. 

Hence, statement (1) alone is not sufficient to answer the question: What is the greatest common divisor of x and y?  

Step 4: Analyze Statement 2

Statement 2 says:  y/6 is an integer.
Now, we know that y is a multiple of 6. However, we don’t know the value of x and y. So, we can’t find the greatest common divisor of the two numbers.

Hence, statement (2) alone is not sufficient to answer the question: What is the greatest common divisor of x and y?  

Step 5: Analyze Both Statements Together (if needed)

Since statement (1) and (2) alone are not sufficient to answer the question, let’s analyse them together.

Statement (1): x = y + 6

Statement (2): y is a multiple of 6.

Combining both statements, we get that x is also a multiple of 6 since x = y + 6.

Now, we know that 6 is a divisor of both the numbers. Let’s check whether it’s the greatest common divisor or not.

Let’s say y is divisible by 7, then x cannot be divisible by 7 since the difference between x and y is 6 only.

The same can be said about all the integers greater than 6. No such integer will be a factor of both x and y. So, the greatest common divisor of x and y is 6.

Thus, both the statements combined can answer the question: What is the greatest common divisor of x and y?

Answer: Option (C)  

Steps 1 & 2: Understand Question and Draw Inferences

Given:   A = m³n²

                        m, n are distinct prime numbers

To find: If 54 is a factor of A

54 = 3³ × 2

Comparing with A = m³n², we can infer that:

If m = 3 and n =2, then 54 will be a factor of A.

For any other values of m and n, 54 will not be a factor of A.

-->  To be able to answer the question, we should be able to determine the values of m and n

Step 3: Analyze Statement 1

1. 25mn is the least common multiple of 15 and 50

 Let’s first find the LCM of 15 and 50:

15 = 3 × 5

50 = 2 × 5²

 To find the LCM, we will pick up the greatest power of each prime factor:

• The power of 2 is 2⁰ in 15 and 2¹ in 50. So, we will pick up 2¹
• The power of 3 is 3¹ in 15 and 3⁰ in 50. So, we will pick up 3¹
• The power of 5 is 5¹ in 15 and 5² in 50. So, we will pick up 5²

LCM = Product of the numbers picked up above

-->  LCM of 15 and 50= 2¹ × 3¹  × 5²                                                                       . . . . Equation ①

Given: LCM of 15 and 50 = 25mn = 5²mn                                                               . . . . Equation ②

 Comparing Equations ① and ②:

 mn =  2× 3 

 -->  (m,n) = (2,3) or (3,2)

 Thus, information given in Statement 1 is not sufficient to arrive at a unique solution

 Step 4: Analyze Statement 2

1. The greatest common factor of 15m³ and 14n² is 6

 6 = 2× 3 

 15m³ = 3  × 5 × m³ 

6 is a factor of 15m³

-->  3  × 5 × m³  is divisible by 2× 3 

-->  m = 2

 14n² = 2  × 7 × n² 

6 is a factor of 14n²

-->  2  × 7 × n²    is divisible by 2× 3 

-->  n = 3

 Thus, we get unique values of m and n.

Since m = 2 and n=3, we can conclude that 54 is not a factor of A.

 -->  Information given in Statement 2 is sufficient to arrive at a unique answer.

 Step 5: Analyze Both Statements Together (if needed)

This step is not required since Step 4 does yield a unique solution

Answer: Option (B)

Step 1: Question statement and Inferences

We are given that positive integers p and q are in the ratio 4: 5. Also, the least common multiple of p and q is given as 800. We have to find the greatest common divisor of the given numbers. 

Since p and q are in the ratio 4: 5, we can write them as: 

p = 4x

q = 5x

Step 2: Finding required values

Given that: 

p = 2² * x

q = 5¹ * x

Least common multiple = 2² * 5¹ * x        (Since the prime numbers that make x will be common in both the numbers)

Greatest common divisor = x                        (Since there is no other common factor in the two numbers)

As we know, the product of least common multiple and greatest common divisor of two numbers is equal to the product of the numbers themselves, we can say:

4x * 5x = 800 * x

20 * x² = 800 * x

So,  x = 40

Step 3: Calculating the final answer

So, the greatest common divisor of p and q will be 40.  

Answer: Option (B)