Lesson: Prime Factorization

Comment on Prime Factorization

would you plz help me with this

If K is an integer and 2 < k < 8, what is the value of k?

1) k is a factor of 30

2) k is a factor of 12.

my answer was c
the prime factorization for 30 is 2-3-5
for 12 is 2-2-3
so just 3 between them so it is c
what is the wrong here?
gmat-admin's picture

Be careful. The question does not say that "k is a PRIME factor of 30." It just says "k is a factor of 30."

Statement 1: The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30.
Since we're told that 2 < k < 8, we can conclude that k = 3, 5 or 6

Statement 2: The factors of 12 are: 1, 2, 3, 4, 6, 12.
Since we're told that 2 < k < 8, we can conclude that k = 3, 4 or 6

The two statements COMBINED
Statement 1 tells us that k = 3, 5 or 6
Statement 2 tells us that k = 3, 4 or 6
Since both statements must be true, we can see that k can still equal 3 or 6.

Answer: E

Hi Brent,

I think the last answer choice has a mistake? "332" has "2" distinct primes.

Which of the following numbers has the greatest number of distinct prime factors?

A. 165
B. 192
C. 228
D. 330
E. 332
This requires us to find the prime factorization of each answer choice

A. 165 = (3)(5)(11) --- 3 distinct prime factors
B. 192 = (2)(2)(2)(2)(2)(2)(3) --- 2 distinct prime factors
C. 228 = (2)(2)(3)(19) --- 3 distinct prime factors
D. 330 = (2)(3)(5)(11) --- 4 distinct prime factors
E. 332 = (2)(2)(83) --- 3 distinct prime factor

Hi Brent,

This was one of the links posted for this vid: https://gmatclub.com/forum/stonecold-s-mock-test-217160.html

Are there any questions from here that you suggest we focus on? Or should we just save the set for further refining post answering the other links?

Many thanks,
gmat-admin's picture

Hi Neel,

I don't see a link to https://gmatclub.com/forum/stonecold-s-mock-test-217160.html from any of the above links.

With the links in the Reinforcement Activities boxes, I suggest that you answer as many as you feel are necessary to get to the level of expertise you need to achieve your target score.

For some students, this will mean answering a handful of questions in the 500-650 range. For others, it will mean answering all of the questions in the 650 range, etc.

Does that help?


Oops! In that case I must have clicked it erroneously

Definitely, thank you for clarifying!


Hi Brent,

Could you please help me with the below question:

Does the integer k have a factor p such that 1 < p <k?

(1) k > 4!
(2) 13! + 2 ≤ k ≤ 13! + 13
gmat-admin's picture

Hi Brent,

Can you please explain this below mentioned question solution in

gmat-admin's picture

Hi Fatima-Zahra,

Here's my full solution: https://gmatclub.com/forum/if-x-is-the-product-of-the-positive-integers-...



Would like to know if my approach is good on the following problem https://gmatclub.com/forum/a-number-is-said-to-be-prime-saturated-if-the-product-of-all-the-diffe-106511.html

Let a,b,c be different prime factors of n. As per question a*b*c < square root of n, if we square both sides we get (a*b*c)^2 < n
Now on to checking numbers:
99= 3*3*11=> (3*11)^2 must be less than 99, which it is not
98=7*7*2 => (7*2)^2 must be less than 98, not true
97 is a prime number 97^2 is more than 97
96= 3*2*2*2*2*2 => (3*2)^2 must be less than 96, 36<96, true.
Answer 96
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-number-is-said-to-be-prime-saturated-if-the...

Your approach is perfect - nice work!


Hi Brent,

Can you help with this problem https://gmatclub.com/forum/if-x-y-and-z-are-integers-and-2-x-5-y-z-0-00064-what-is-the-188182.html

I am completely stuck. I represented 0.00064 as 64*10^-5, which can be represented as 2^6*10^-5, I do not know what to do from here. Or maybe the whole approach is wrong?
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-y-and-z-are-integers-and-2-x-5-y-z-0-000...

That's a good start.
The trick now is to recognize that there are many different ways to rewrite (2^6)(10^-5)

For example, we know that (2^-5)(5^-5) = 10^-5
So, we can write: (2^6)(10^-5) = (2^6)(2^-5)(5^-5)

Or we can combine the two powers of 2 to get: (2^6)(2^-5)(5^-5) = (2^1)(5^-5)

And so on.

Here's my full solution: https://gmatclub.com/forum/if-x-y-and-z-are-integers-and-2-x-5-y-z-0-000...


Hi Brent
could you please help me
from times to times in this topic i found a question like:
"how many different prime factors does n have?"
"n is divisible by how many positive integers?"
the problem is that it's not clear for me what I need to find. Should I found how many different factors N has OR should I found how many different numbers/quantity factors N has.
examples below
for me both of these questions sound the same, but they are different. how to dedicate\understand those differences?

gmat-admin's picture

Question links:

If we're asked to find the number of DIFFERENT PRIME factors, we are counting ONLY prime factors, and we cannot count repeated primes more than once.
For example, 18 = (2)(3)(3)
So, 18 has TWO DIFFERENT prime factors: 2 and 3

Next, asking "n is divisible by how many positive integers?" is the same as asking "How many POSITIVE FACTORS does n have?"
In this case, we are counting ALL factors (prime and not prime)

Example: How many POSITIVE FACTORS does 18 have?
Factors of 18: 1, 2, 3, 6, 9, 12
So, 18 has 6 positive factors.

Does that help?


Question link: https://gmatclub.com/forum/if-60-is-written-out-as-an-integer-with-how-many-101752.html

Hi Brent, can you please share your solution to this problem?

gmat-admin's picture

Hi Brent, can you please solve question DS00340 - Q390 of OG 2019?

(A school will assign each student in a group of n students to one of m classrooms. If 3<m<13<n, is it possible to assign each of n students to m classrooms so that each class has same number of students?)

According to me the answer is D. But correct answer is B, which I couldn't understand from OG's solution.

gmat-admin's picture


Approach plz
gmat-admin's picture

If x is a prime number greater than 2, which of the following could be a prime number?

(A) x^2
(B) x/2
(C) 3x
(D) x−4
(E) x^2 + 1

Check the answer choices.....

Answer: D

But here what if 13 -4 =9 pls explain 9 is not a prime
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-is-a-prime-number-greater-than-2-which-o...

The key word here is COULD. So, we need just ONE instance in which x - 4 is prime in order for D to be correct.
If x = 11 (which is prime), then x - 4 is prime.
So, x - 4 CAN be prime.

Your example (using x = 13) simply shows that x - 4 isn't always prime (but the correct answer is still D, since x - 4 CAN be prime)

If, on the other hand, the question asked, "...which of the following MUST be a prime number?", then answer choice D would not be correct.

How many prime numbers between 1 and 100 are factors of 7,150?

factors of 7150 ...whats the shortest trick pls explain

Hi, is there a less time consuming way to solve this question:

If 60! is written out as an integer, with how many consecutive 0’s will that integer end?


gmat-admin's picture

The key concept here is: For every pair of one 2 and one 5, we get a product of 10, which accounts for one zero at the end of the integer.

Since there are A LOT more 2's than 5's "hiding" in 60!, it all comes down to counting the number of 5's hiding in 60!
We have 5, 10, 15, 20, 25 (there are two 5's hiding in 25), etc.
The entire process shouldn't take longer than 1 minute.
Here's my full solution: https://gmatclub.com/forum/if-60-is-written-out-as-an-integer-with-how-m...

Hi Brent, Can you explain me this question?

gmat-admin's picture

Thank you

Hi Brent, could you please help me explain how different are these 2 questions?

1) https://gmatclub.com/forum/in-n-is-a-positive-integer-and-14n-60-is-an-integer-then-n-has-how-ma-224145.html

2) https://gmatclub.com/forum/if-n-is-a-positive-integer-less-than-200-and-14n-60-is-an-100763.html

I am struggling to understand the difference as in Question 2, it says " In N is a positive integer less than 200 ", here N not necessarily has to be 30 right, it can also be 40 as well.
gmat-admin's picture

Question #1: https://gmatclub.com/forum/in-n-is-a-positive-integer-and-14n-60-is-an-i...
Question #2: https://gmatclub.com/forum/if-n-is-a-positive-integer-less-than-200-and-...

Question #2 says "N is a positive integer less than 200"
Given: 14N/60 is an integer
Simplify to get: 7N/30 is an integer.
This means N must be divisible by 30.
So, N can equal 30, 60, 90, 120, etc.
In fact, N can be any multiple of 30. So, the possible values of N range from 30 all the way to 180 (since N is less than 200)

The question asks "N has how many DIFFERENT positive prime factors?"
All multiples of 30 from 30 to 180 have the same number of DIFFERENT prime factors. That is, for each possible value of N, the three different prime factors are 2, 3 and 5.

In question #1, the value of N is not limited.
So, N can be ANY multiple of 30.
If N = 30, then N has 3 different prime factors: 2, 3 and 5
If N = 210, then N has 4 different prime factors: 2, 3, 5 and 7
If N = 2310, then N has 5 different prime factors: 2, 3, 5, 7 and 11

Does that help?

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