If x < 20, How many distinct factors does odd number x have? 1) 16x

IMO distinct factors are just the last prime numbers and not any other possible factor.
Here only 3&1 are the prime numbers but 9 is not


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I read somewhere as distinct factors=distinct prime factors. And factors are different from distinct factors


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x is odd number less than 20, so we have the entire possible values of x as:- 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

(1) 24 = 2^3 * 3 and 16 = 2^4.
So 16 has already the required number of 2's but for 16x to be divisible by 24, there needs to be a factor of '3' in the numerator which is coming from x. This means x is divisible by 3. So now the possible values of x are = 3, 9, 15. '3' has two distinct factors (1, 3); 9 has three distinct factors (1, 3, 9). So not sufficient.

(2) 14 = 2*7 and 15 = 3*5.
So 14 does not have either a 3 or a 5. And since 14x is NOT divisible by 15, it means x does not contain both 3 & 5 as factors, because in that case it will become divisible by 15. (Please note that x can contain one factor out of 3 or 5 but not both). So as per this the possible values of x are = 1, 3, 5, 7, 9, 11, 13, 17, 19 (only 15 is excluded). So not sufficient.

Combining both statements, x can still be either '3' or '9'. And while '3' has two distinct factors (1, 3); 9 has three distinct factors (1, 3, 9). So still the data is not sufficient. Answer should be E.

(would request you to check the OA, which is mentioned as C)

Is distinct factors of 9, 1& 3 only? Could someone please explain. I think it should be 1,3&9 and OA should be C.

A factor is a positive divisor of an integer, so a positive integer which divides another integer without a reminder. The (distinct) factors of 9 are 1, 3, and 9.



Almost all official question involving factors are about distinct positive prime or non-prime factors. There is one question (https://gmatclub.com/forum/the-prime-su ... 67088.html) which explicitly mentions "...prime factors of n, including repetitions", when it wants non-distinct primes to be counted as well.

A prime factor is a factor of an integer which is a prime. For example, the factors of 12 are: 1, 2, 3, 4, 6, and 12 and PRIME factors of 12 are 2 and 3 only.

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Bunuel,

Please clarify the controversy over this Topic. What shall be the OA? C or E?

In its current form the answer should be E. x can be any odd multiple of 3, which is not a multiple of 5 less than 20: 9, 3, -3, -9, -21, -27, -33, ...

GMATinsight could you please revise the question/OA? Thank you.

Thank for the clarification Bunuel

While mathematically, you may be correct. But in GMAT, when question asks about distinct factor, it is positive or at least mention positive. Can you cite any official question for factors of negative numbers?

Thanks

GMAT divisibility/remainder questions are restricted to positive integers only, which means that a GMAT question will mention in advance that all variables are positive integers (for divisibility/remainder questions). But if a question does not mention this, then a variable can without a doubt be negative. For this particular question we are not told that x is a positive integer, so x could be a negative integer. As far as factors go, then they are always by default positive integers.

E is correct.
Both (1) and (2) can have infinite solutions, because - value also can included.

Yeah. Became clear now. Thanks


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Hi

No we cannot include negative values. The question asks about distinct factors and 'factors' on GMAT are always taken as positive.

Combining (1) and (2), the reason the answer is E and not C is that after combining the two statements, we come down to two numbers - 3&9; and while '3' has 2 distinct factors, '9' has three distinct factors. Since there is no unique answer, we mark the answer as E.

A factor cannot be negative but x itself can be. x can be any odd multiple of 3, which is not a multiple of 5 less than 20: 9, 3, -3, -9, -21, -27, -33, ...

Hi

No we cannot include negative values. The question asks about distinct factors and 'factors' on GMAT are always taken as positive.

Combining (1) and (2), the reason the answer is E and not C is that after combining the two statements, we come down to two numbers - 3&9; and while '3' has 2 distinct factors, '9' has three distinct factors. Since there is no unique answer, we mark the answer as E.[/quote]

A factor cannot be negative but x itself can be. x can be any odd multiple of 3, which is not a multiple of 5 less than 20: 9, 3, -3, -9, -21, -27, -33, ...[/quote]


Yes, x can be negative I get it. Thanks for clarification.