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

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

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

1) 16x is divisible by 24

2) 14x is not divisible by 15


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Originally posted by GMATinsight on 22 Jan 2018, 05:33.
Last edited by GMATinsight on 23 Jan 2018, 03:58, edited 1 time in total.
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 22 Jan 2018, 08:57
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C

From question stem we can conclude that x is an odd number. No other info.

From st1 we have 16x/24 ie 2x/3 holds good. Now x can be 3, 9, 15. Distinct factors of 3,9 are 3&1 but for 15 it’s 3, 5 &1. Ambiguous values. Not sufficient.

From st2 we can decide x can be any odd number < 20 other than 15. Multiple values. Hence insufficient.

Combining value of x as 15 is ruled out. Only 3&9 exist with distinct factors as 3,1. Only two factors. Hence sufficient
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 22 Jan 2018, 20:30
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How come 9 has only two distinct factors?

Is n't factors of 9 : {1,3,9} distinct ?

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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 22 Jan 2018, 20:32
hellosanthosh2k2 wrote:
How come 9 has only two distinct factors?

Is n't factors of 9 : {1,3,9} distinct ?

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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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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 22 Jan 2018, 20:47
hellosanthosh2k2 wrote:
I think question should have mentioned as prime factor explicitly. But we can't rule out 9 as not as factor unless I am misunderstanding.

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


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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 22 Jan 2018, 22:49
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Edit:
GMATinsight wrote:
If x < 20, How many distinct factors does odd number x have?

1) 16x is divisible by 24

2) 14x is not divisible by 15


Source: http://www.GMATinsight.com


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)
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 22 Jan 2018, 23:59
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.
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 00:11
sghoshgt wrote:
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.

Sasindran wrote:
hellosanthosh2k2 wrote:
I think question should have mentioned as prime factor explicitly. But we can't rule out 9 as not as factor unless I am misunderstanding.

Posted from my mobile device


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


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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.

2. Properties of Integers




5. Divisibility/Multiples/Factors



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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 00:17
Bunuel,

Please clarify the controversy over this Topic. What shall be the OA? C or E?
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 00:28
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Sasindran wrote:
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.
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 04:14
Thank for the clarification Bunuel
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 09:14
Bunuel wrote:
Sasindran wrote:
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.



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
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 09:22
Mo2men wrote:
Bunuel wrote:
Sasindran wrote:
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.



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.
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 19:39
Sasindran wrote:
C

From question stem we can conclude that x is an odd number. No other info.

From st1 we have 16x/24 ie 2x/3 holds good. Now x can be 3, 9, 15. Distinct factors of 3,9 are 3&1 but for 15 it’s 3, 5 &1. Ambiguous values. Not sufficient.

From st2 we can decide x can be any odd number < 20 other than 15. Multiple values. Hence insufficient.

Combining value of x as 15 is ruled out. Only 3&9 exist with distinct factors as 3,1. Only two factors. Hence sufficient


E is correct.
Both (1) and (2) can have infinite solutions, because - value also can included.
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 20:11
HG0815 wrote:
Sasindran wrote:
C

From question stem we can conclude that x is an odd number. No other info.

From st1 we have 16x/24 ie 2x/3 holds good. Now x can be 3, 9, 15. Distinct factors of 3,9 are 3&1 but for 15 it’s 3, 5 &1. Ambiguous values. Not sufficient.

From st2 we can decide x can be any odd number < 20 other than 15. Multiple values. Hence insufficient.

Combining value of x as 15 is ruled out. Only 3&9 exist with distinct factors as 3,1. Only two factors. Hence sufficient


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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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 22:43
HG0815 wrote:
Sasindran wrote:
C

From question stem we can conclude that x is an odd number. No other info.

From st1 we have 16x/24 ie 2x/3 holds good. Now x can be 3, 9, 15. Distinct factors of 3,9 are 3&1 but for 15 it’s 3, 5 &1. Ambiguous values. Not sufficient.

From st2 we can decide x can be any odd number < 20 other than 15. Multiple values. Hence insufficient.

Combining value of x as 15 is ruled out. Only 3&9 exist with distinct factors as 3,1. Only two factors. Hence sufficient


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


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.
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 23:13
amanvermagmat wrote:
HG0815 wrote:
Sasindran wrote:
C

From question stem we can conclude that x is an odd number. No other info.

From st1 we have 16x/24 ie 2x/3 holds good. Now x can be 3, 9, 15. Distinct factors of 3,9 are 3&1 but for 15 it’s 3, 5 &1. Ambiguous values. Not sufficient.

From st2 we can decide x can be any odd number < 20 other than 15. Multiple values. Hence insufficient.

Combining value of x as 15 is ruled out. Only 3&9 exist with distinct factors as 3,1. Only two factors. Hence sufficient


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


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, ...
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Re: If x < 20, How many distinct factors does odd number x have? 1) 16x [#permalink]

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New post 23 Jan 2018, 23:19
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.
Re: If x < 20, How many distinct factors does odd number x have? 1) 16x   [#permalink] 23 Jan 2018, 23:19
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