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# If a,b,k, and m are positive integers, is a^k a factor of

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If a,b,k, and m are positive integers, is a^k a factor of [#permalink]

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07 Apr 2011, 23:32
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56% (00:23) correct 44% (01:01) wrong based on 16 sessions

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If a,b,k, and m are positive integers, is a^k a factor of b^m?

1) a is a factor of b

2) k ≤ m

I don't know OA.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-a-b-k-and ... 88503.html
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Kudos [?]: 1135 [0], given: 39

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Joined: 14 Feb 2011
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Re: If a,b,k, and m are positive integers, is a^k a factor of [#permalink]

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07 Apr 2011, 23:54
Baten80 wrote:
If a,b,k, and m are positive integers, is a^k a factor of b^m?

1) a is a factor of b

2) k ≤ m

I don't know OA.

we need to find out if $$\frac{b^m}{a^k}$$ is an integer.

Statement 1 says a is a factor of b so $$b = na$$ where n is an integer.

So $$\frac{b^m}{a^k}$$ = $$\frac{(na)^m}{a^k}$$ = $$\frac{(n^m*a^m)}{a^k}$$

This will be integer if $$a^m \geq a^k$$ which will happen only if $$k \leq m$$which is not given in statement 1, so insufficient

Statement 2 doesnt say anything about a and b so insufficient

Combining statement 1 and 2 we can answer the question in affirmative, hence sufficient and Answer is C.

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Re: If a,b,k, and m are positive integers, is a^k a factor of [#permalink]

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08 Apr 2011, 00:33
Is b^m = r *a^k ? (where p is as an integer)

(1) b = p * a (where p is as an integer)

so b^k = p^k * a^k

but we don't know the relationship between m and k, so insufficient.

e.g. a = 3, b = 6

so 6^2/3^3 is not an integer.

(2) No information about a and b, so insufficient.

(1) and (2) => b^m = p^k * a^k or, b^m/a^k = p^k (an integer), so sufficient

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Kudos [?]: 592 [0], given: 36

Director
Joined: 01 Feb 2011
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Kudos [?]: 144 [0], given: 42

Re: If a,b,k, and m are positive integers, is a^k a factor of [#permalink]

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08 Apr 2011, 18:22

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Kudos [?]: 144 [0], given: 42

Manager
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Re: If a,b,k, and m are positive integers, is a^k a factor of [#permalink]

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08 Apr 2011, 20:22
Take a = 2, b = 6
=>2^k and 6^m
=>2^k and 2^m x 3^m
If k<=m, for any positive value of k and m, 2^k is always a factor of 2^m
So C
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Kudos [?]: 37 [0], given: 4

Intern
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Re: If a,b,k, and m are positive integers, is a^k a factor of [#permalink]

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12 Sep 2016, 11:48
Baten80 wrote:
If a,b,k, and m are positive integers, is a^k a factor of b^m?

1) a is a factor of b

2) k ≤ m

I don't know OA.

1) b^m = a^k * n. Since a is factor of B, let a =2, b=8
so 2^3m = 2^k * n. 2^k can only be a factor of 2^3m if the exponents of 2^k is less than 2^3m for all k and m. So not sufficient

2)k <= m, not sufficient because a and b relationship not given. If a = 3, b = 5, can be yes or no for a^k a factor of b^m

1+2) given a is a factor of b and the exponent of a will be smaller than or equal to the exponent of b, a^k is then a factor of b^m. C

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Kudos [?]: 129127 [0], given: 12194

Re: If a,b,k, and m are positive integers, is a^k a factor of [#permalink]

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14 Aug 2017, 06:25
Baten80 wrote:
If a,b,k, and m are positive integers, is a^k a factor of b^m?

1) a is a factor of b

2) k ≤ m

I don't know OA.

Question: is $$a^k$$ a factor of $$b^m$$ --> $$a^kx=b^m$$, where $$x$$ is an integer? --> $$x=\frac{b^m}{a^k}$$. So basically the question is: Is $$x$$ an integer $$>0$$?

(1) $$a$$ is a factor of $$b$$ --> $$ay=b$$ --> $$x=\frac{a^my^m}{a^k}$$ --> $$x=a^{m-k}y^m$$. Now if $$m<k$$ and $$a$$ is not a factor of $$y$$, then $$x$$ will not be an integer. Not sufficient.

Or even without any algebra: if a and b are equal to say 3 and m<k (there are less b's than a's) then a^k won't be a factor of b^m. Though if k<=m then even if a and b are not equal still a^k will be a factor of b^m as there will be enough b's for a's.

(2) $$k\leq{m}$$, not sufficient on it's own.

(1)+(2) $$x=a^{m-k}y^m$$ and $$k<m$$, hence $$x$$ is an integer. Sufficient. (Or again as there are more b's then a's (enough b's for a) then a^k is a factor of b^m, for example (bbb)/(aa))

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-a-b-k-and ... 88503.html
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Kudos [?]: 129127 [0], given: 12194

Re: If a,b,k, and m are positive integers, is a^k a factor of   [#permalink] 14 Aug 2017, 06:25
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