A positive integer n has 3 prime factors. Is n odd?
(1) The total number of factors of n is odd.
(2) The prime factors of n when arranged in ascending order form an evenly spaced set.
Step 1 & 2: Understand Question and Draw Inference
Given:
To find: is n odd?
To determine the evenodd nature of n, we need to see if 2 is one of the prime
factors (P_{1} or P_{2} or P_{3} ) of n
Step 3 : Analyze Statement 1 independent
1. The total number of factors of n is odd
The statement tells us that n is a perfect square, but it does not tell us if 2 is one of the prime factors of n.
Hence, statement1 is insufficient to answer the question.
Step 4 : Analyze Statement 2 independent
2. The prime factors of n when arranged in ascending order form an evenly spaced set.
As P_{1} , P_{2} , P_{3} form an evenly spaced set, we can write {P_{1} , P_{2} , P_{3}} = { P_{1},P_{1} +d, P_{1} +2d} assuming P_{1} < P_{2} < P_{3}
So, P + P = 2P i.e. P + P is even.
This would be possible only if P_{1} and P_{3} both are odd. P_{1} and P_{3} both can’t be even as there is only one even prime number possible.
As P_{2} lies between P_{1} and P_{3}, P_{2} also has to be odd as there is no even prime number which lies between two odd prime numbers.
Hence none of P_{1} , P_{2}, or P_{3} is 2. Thus n is definitely odd.
So, statement2 is sufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
Since, we have a unique answer from step 4, this step is not required.
Answer: B
The product of the largest two factors of a positive integer n is 16875. What is the difference between the largest positive integer that divides √n and the smallest odd factor greater than 1 of √n?
Given:
To Find: Difference between the largest positive factor that divides √n and the smallest odd factor of √n greater than 1.
Approach:
c. For finding the smallest prime factor of n, we need to find the smallest prime factor of 16875.
Working out:
c. As the smallest prime factor of 16875 is 3, the second largest factor of n = n/3 =b
2. So, √n = 15
3. Largest positive factor that divides √n = √n = 15
4. Smallest odd factor of √n greater than 1 = smallest odd prime factor of √n = 3
5. So, Largest positive factor that divides √n  Smallest odd factor of √n greater than 1 = 15 – 3 = 12
Answer B
For a positive integer n, the function @n represents the product of all even numbers that lie between n and 2n, exclusive. For
example, @6 = 8*10. If k is a positive integer and the greates prime factor of @k is 17, which of the following cannot be the
value of k?
I. 9
II. 18
III. 32
Given:
To Find: Which of the given 3 values doesn’t satisfy the given information about k
Approach:
prime factor:
a. If the answer is YES, then that value of k is possible
b. If the answer is NO, then that value of k is not possible
Working out:
Looking at the answer choices, we see that the correct answer is Option D From the table, it’s clear that out of the 3 given values, Options I and III (9 and 32) are not possible (Remember that the question is asking which values are not possible)
For any integer P greater than 1, P! denotes the product of all the integers from 1 to P, inclusive. The value of 11! is closest to
which of the following powers of 10?
Given:
To Find: Which power of 10 is 11! The closest to?
Approach:
Working out:
Looking at the answer choices, we see that the correct answer is Option B
Does the positive integer M have greater than 4 factors?
(1) M/8 is an integer
(2) M/20 is an integer
Step 1 & 2: Understand Question and Draw Inference
Given: Positive integer M
To find:
Step 3 : Analyze Statement 1 independent
M = 2^{3+a} *P_{2}^{b} * P_{3}^{c} . . . where a, b, c . . . are nonnegative integers
Thus, Statement 1 is not sufficient to arrive at a definite answer to the question.
Step 4 : Analyze Statement 2 independent
Thus, Statement 2 is sufficient to provide us a definite answer to the question.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
p is a positive integer greater than 100. Is p divisible by 36?
(1) There are 8 positive integers, including 1 and p, which divide p.
(2) The highest positive integer less than or equal to 100 that divides p is 75.
Step 1 & 2: Understand Question and Draw Inference
The question tells us about a positive integer p greater than 100. It asks us to find if p is divisible by 36
Let’s first analyze the information given in the question statement.
36 = 2^{2} * 3^{2} . For p to be divisible by 36, p should have at least 2 and 3 in it.
Also, since 36 has (2+1) (2+1) = 3 *3 = 9 factors, p should have a minimum of 9 factors.
With the above understanding, let’s see the information provided in the statements.
Step 3 : Analyze Statement 1 independent
1. There are 8 positive integers, including 1 and p, which divide p Statement1 tells us that p has a total of 8 number of factors. We know that for p to be divisible by 36, it should have a minimum of 9 factors. As p has less than 9 factors, p is definitely not divisible by 36.
Hence statement1 is sufficient to answer the question.
Step 4 : Analyze Statement 2 independent
2. The highest positive integer less than or equal to 100 that divides p is 75.
Statement2 tells us that 75 is the highest integer less than equal to 100 which divides p.
75 = 5^{2}* 3. Had p been divisible by 36, p should also have contained at least 2 and 3 .
So, p would have at least 2 , 3 and 5 in it. With this combination of the prime factors and their powers, the highest integer less than equal to 100 which divides p should have been 5 * 2 = 100. As 75 is the highest integer less than equal to 100 which divides p, we can say that p does not contain 2^{2} . So p is definitely not divisible by 36.
Hence statement2 is sufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
Since we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
Can the total number of integers that divide x be expressed in the form of 2k + 1, where k is a positive integer?
(1) √12x is an integer
(2) The product of √x and √y is an integer, where the total number of factors of y/3 is odd.
Step 1 & 2: Understand Question and Draw Inference
To Find: Is the number of factors of x odd?
Step 3 : Analyze Statement 1 independent
is an integer. For √3x to be an integer,
x should contain an odd power of 3.
Now, if 3 occurs odd number of times in x, x can’t be a perfect square.
Hence statement1 is sufficient to answer the question.
Step 4 : Analyze Statement 2 independent
2. The product of √x and √y is an integer, where the total number of factors of y/3 is odd.
Statement2 tells us that the total number of factors of y/3 is odd i.e. y/3 is a perfect square. Let’s assume y/3 = 3z^{2}. Since z^{2} is always nonnegative, y will also be a nonnegative integer
So, we know that is an integer i.e. z√3x is an integer.
For √3x to be an integer, x should contain an odd power of 3. If 3 occurs odd
number of times in x, x can’t be a perfect square
Hence statement2 is sufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
Since we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
If m has the smallest prime number as its only prime factor, is ∛m an integer?
(1) m^{2} is divisible by 32
(2) √m is divisible by 4
Step 1 & 2: Understand Question and Draw Inference
Given:
To find:
So, in order to answer the question, we need to find if n is a multiple of 3 or not
Step 3 : Analyze Statement 1 independent
Step 4 : Analyze Statement 2 independent
Since √m is divisible by 4, √m must be greater than or equal to 4
Statement 2 alone is not sufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
We’ve still not been able to determine if n is a multiple of 3.
So, even the two statements together are not sufficient to answer the question.
Answer: Option E
For any integer P greater than 1, P! denotes the product of all the integers from 1 to P, inclusive. In how many ways can 9! be
expressed as a product of 2 positive integers?
Given:
P > 1
P! = 1 x 2 x 3…x P
To Find
# of ways
in which 9! can be written
as a x b , where a, b are integers > 0
Approach:
1. Note the nature of numbers a,b in the product . a x b.
a,b are both factors of 9!.
Let’s take a simple example to understand the above application. Let’s take the number 6.
Its factors are – 1, 2, 3, 6  i.e. 4 factors
It can be expressed as product of two numbers as follows
1 x 6
2 x 3
Working out:
9! can be expressed as a product of two positive numbers in 80 ways. Choice D is the correct answer.
What percentage of odd prime numbers lying between 1 and 30 divide 7,700 completely?
Given:
Approach:
2. For finding the prime numbers that divide 7700, we need to find the prime factors of 7700, which will tell us the odd prime numbers that
divide 7700
3. Then we will find the number of odd prime numbers lying between 1 and 30.
4. Using (1) and (2), we can find the percentage of odd prime numbers lying between 1 and 30, that divide 7700
Working out:
2. Number of odd prime numbers lying between 1 and 30 = {3, 5, 7, 11, 13, 17, 19, 23, 29} = 9 odd prime numbers.
Answer : D
If the dimensions of a rectangle (in inches) is equal to a prime number that lies between 55 and 65, exclusive, which of the
following statements must be true?
I. There is only one other rectangle that has integral dimensions and the same area as the given rectangle
II. There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangle
III. The area of the given rectangle is 3599 square inches.
Given:
So, it would be wrong to assume so.
To Find: Which of the 3 given statements must be true?
Approach:
Working out:
In how many ways can 59*59 be expressed as a product of 2 integers?
Looking at the answer choices, we see that the correct answer is Option A
If k is a positive integer and 6k^{3} is divisible by 2500, then which of the following statements must be true?
I. k is divisible by 50
II. 20 is a factor of k
III. k^{5} is divisible by 64
Given:
To Find: Which of the 3 statements must be true?
Approach:
Working out:
Since a is an integer, minimum possible value of a = 2
Since b is an integer, minimum possible value of b = 1
Therefore, k = 2*5^{2}* P_{3}^{c} *P_{4}^{d} . . .
Looking at the answer choices, we see that the correct answer is Option D
A class of students appeared in two different tests of quantitative ability both of which were scored out of 100. The scores of each
student in each test could be expressed as a product of two distinct prime numbers. If a student’s scores in the two tests did not have
any common prime factors, what is the maximum possible difference between his two scores?
Given:
To Find: Maximum possible difference between the scores of a student in two tests
Approach:
Working out:
Answer: D
A positive integer n has more than 1 distinct prime factor. Is the product of the prime factors of n less than n?
(1) The number of distinct factors of n, including n, is odd.
(2) n can be expressed as a multiple of 4.
Step 1 & 2: Understand Question and Draw Inference
Step 3 : Analyze Statement 1 independent
1. The number of distinct factors of n, including n, is odd.
Now, we don’t need to analyze if the power of other prime factors of n is greater than 1, because as we have analyzed in steps 1&2 that even if one of the prime factors of n has a power greater than 1, then
However, the other prime factors of n will also have an even power greater than 1.
So, b, c …. > 1
Step 4 : Analyze Statement 2 independent
2. n can be expressed as a multiple of 4.
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
For any integer P greater than 1, P! denotes the product of all the integers from 1 to P, inclusive. If 10! Is divisible by 10080h and h
is a perfect square, what is the greatest possible value of h?
Given:
To Find: The greatest possible value of h
Approach:
2 So, we will first find the value of
3. Then, we will prime factorize this value to determine the greatest perfect square that divides this value.
Working out:
How many odd positive integers divide the positive integer n completely?
(1) 16 is the highest power of 2 that divides n
(2) n has a total of 68 factors and 3 prime factors.
Step 1 & 2: Understand Question and Draw Inference
To Find: Number of odd positive integers divide n?
Step 3 : Analyze Statement 1 independent
Statement1 tells us that the prime number 2 comes 16 times in n.
So, for n = P_{1}^{a} * P_{2}^{b} * P_{3}^{c} ….we know that P_{1} = 2 and a = 16.
However it does not tell us anything about the powers of odd prime factors of n.
Hence statement1 is insufficient to answer the question.
Step 4 : Analyze Statement 2 independent
2. n has a total of 68 factors and 3 prime factors.
Statement2 tells us that n has a total of 68 factors and 3 prime factors.
So, for n = P_{1}^{a} * P_{2}^{b} * P_{3}^{c} we know that (a+1) (b+1) (c+1) = 68.
Now, let’s find the number of ways in which 68 can be expressed as a product of 3 integers > 1.
68 = 2 * 2 * 17.
So, (a, b, c) = (1, 1, 16). However, we do not know:
If 2 is a prime factor of n AND
Even if 2 is a prime factor of n, the power of 2 in n.
So, we cannot in any way find out the number of odd positive integers that divide n.
Hence statement2 is insufficient to answer the question.
Step 5: Analyze Both Statements Together (if needed)
Since we do not have a unique answer from either step 3 or 4, we need to
analyze both the steps together.
Step3 tells us that P_{1} = 2 and a = 16.
Step 4 tells us that n = P_{1}^{a} * P_{2}^{b} * P_{3}^{c} and (a+1) (b+1) (c+1) = 68.
Combining both the steps, we can say that (b+1) (c+1) = 4 i.e. there are a total
of 4 odd factors of n.
Hence statements 1 and 2 together are sufficient to answer the question.
Answer : C
For any integer x greater than 1, x! denotes the product of all integers from 1 to x, inclusive. What is the hundredthousands
digit of 25!?
Given:
To Find:: Hundredthousand digit of 25!
Approach:
Working out:
3. As 25! has 2^{22} and 5^{6} in it, 25! will have 10 in it
4. So, 25! will have 6 zeroes at the end. Hence, the hundredthousands digit will be 0.
Answer A
What is the number of positive integers that divide k but do not divide k, where k is a positive integer?
(1) k^{2 }has a total of 13 factors
(2) √k has a total of 4 factors
Step 1 & 2: Understand Question and Draw Inference
To Find: number of positive integers that divide k^{2} but do not divide k
Step 3 : Analyze Statement 1 independent
1. k^{2} has a total of 13 factors
Hence, there are a total of 6 positive integers that divide k but do not divide k. Sufficient to answer
Step 4 : Analyze Statement 2 independent
1. √k has a total of 4 factors
As we do not have a unique answer, this statement is insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step3, this step is not required
Answer A
For any integer P greater than 1, P! denotes the product of all the integers from 1 to P, inclusive. What is the greatest integer m for
which 45m is a factor of 48!?
Given:
To Find: The greatest integer m for which 45^{m} is a factor of 48!
Approach:
45^{m} = (3^{2} *5)^{m} = 3^{2m} *5^{m}
So, 45^{m} is a factor of 48! only if:
a. 3^{2m} is a factor of 48! AND
b. 5^{m} is a factor of 48!
We need to find the greatest value of m that satisfies this constraint
2. We’ll find the greatest value of m for which 3^{2m} is a factor of 48! Let this value be x
3. Next, we’ll find the greatest value of m for which 5^{m} is a factor of 48! Let this value be y
4. The greatest value of m for which 3^{2m} and 5^{m} are both factors of 48! will be equal to the lesser value between x and y
Working out:
Important Takeaway:
We can summarize the process we used above to find the power of 3 in 48!, in the following formula:
(Power of 3 in 48!)
= (Number of multiples of 3 in 48!) + (Number of multiples of 3^{2} in 48!) + (Number of multiples of 3^{3} in 48!)
Good so far?
Now comes the interesting point:
We can follow this same process to find the power of any prime number X in P! where P is any positive integer
The general formula to summarize the above process will be:
(Power of primefactor X in P!)
= (Number of multiples of X in P!) + (Number of multiples of X^{2} in P!) + (Number of multiples of X^{3} in P!) + ...
Looking at the answer choices, we see that the correct answer is Option D
If z is the product of integers from 1 to 51, inclusive, what is the greatest value of k+ m, where k and m are integers greater than 1,
for which 45^{k} * 32^{m} is a factor of z?
Given:
To Find: Greatest value of k + m , for which 45 * 32 is a factor of z ?
Approach:
Working out:
4. Next, we need to find the number of times, 2, 3 and 5 occur in z = 1*2*3…….51
a. Let’s take a simple example to understand how we can find this. Consider a number p = 1*2*3*4. We need to find the power of 2 in p.
b. Now, powers of 2 will occur in multiples of 2. So, we should first find the multiples of 2 in p. They are 2 and 4. However all multiples of 2 will not contain only 2^{1} . Some of them may contain higher powers of 2. For example, here 4 contain 2^{2} .
c. So, when we are finding multiples of 2, we should also find the multiples of higher powers of 2 in p.
d. Hence, 2 will occur 2 + 1 = 3 times in p.
5. Using the same analogy, let’s find out the powers of 2, 3 and 5 in z
9. Using points (6.c) and (7.c), we can say that maximum value of k = 11
10. Hence, maximum (k + m ) = 11 + 9 = 20
Answer B
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