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

Question

(A) 75
(B) 120
(C) 144
(D) 210
(E) 750

Aditi31 · Accepted Answer

I got confused about whether the question is asking about greatest distinct prime factors or greatest prime factors in general( in count or including the recurring primes). It would be helpful if you could share some light on this.

Gmt123 · Answer

1. 75 - 5,3
2. 120 - 2,3,5
3. 144 - 2,3
4. 210 - 2,3,5,7
5. 750 - 2,3,5

210 has 4 prime factors

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Japji13 · Answer

suppose 2 occur twice first time it will be prime 2^1 . 
 2nd time it multiple of 2 (2^2 =4 ) which is not prime number 
hope it helps

GurpreetMahli · Answer

Bunuel  Does the question stem need to clarify whether it is asking for 'distinct prime factors' or 'total prime factors' (including repeats)?

Bunuel · Answer

No. If a question asks “how many prime factors does 12 have?”, the answer is 2, not 3. The prime factors are 2 and 3.

So “prime factors” and “distinct prime factors” mean the same thing here. You do not count the factor 2 twice just because it appears twice in the prime factorization.

If a question wants repeated prime factors to be counted, it needs to state that explicitly.

egmat · Answer

Hi GurpreetMahli,

Great catch, and you're right to press on this, because here the two readings actually give  different answers . This isn't a case where it doesn't matter.

On a GMAT question, the standard convention is that  "prime factors" means the  distinct  prime numbers that divide the value  - each prime counted once, no matter how many times it appears in the factorization. Under that reading:

-  210 = 2 × 3 × 5 × 7  - four distinct primes - the winner, which is why the answer is  D .

So the answer key itself confirms the intended meaning is  distinct  primes.

Why the wording matters here

Watch what happens if you instead counted  every  prime including repeats:

- (A) 75 = 3 × 52 -  3  total
- (B) 120 = 23 × 3 × 5 -  5  total
- (C) 144 = 24 × 32 -  6  total
- (D) 210 = 2 × 3 × 5 × 7 -  4  total
- (E) 750 = 2 × 3 × 53 -  5  total

With repetition,  144  would win with  6  - not  210 . So your instinct was sharp: the two interpretations genuinely diverge, and the OA of D tells us the question means  distinct  prime factors.

Japji13's note above is really getting at this same idea: once you've used the prime  2 , higher powers like  4 ,  8 ,  16  aren't  new  primes - they're just more copies of  2 . Distinct-prime counting ignores those extra copies.

Quick rule to carry forward:  unless a question explicitly says "counted with repetition" or "total factors," read  "number of prime factors" as distinct primes . That's the safe default on this exam.

Answer: D

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

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

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