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

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Joined: 02 Sep 2009
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If a, b, k, and m are positive integers, is a^k a factor of b^m?  [#permalink]

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13 Jun 2019, 04:22
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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.

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

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13 Jun 2019, 04:27
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If a, b, k, and m are positive integers, is a^k a factor of b^m?

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 greater than 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 ≤ 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))

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

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13 Jun 2019, 10:07
1
Bunuel 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.

good question
#1
a is a factor of b
a = 3 and b = 6
so
if k>m ; we get no and when k<m we get yes
test with k=1 and m= 1 and k=3 & m=1
insufficient
#2
k<=m
again relation of a & b is not clear so it can be yes and no
a=3 and b = 6 or a =2 and b = 3 so values of m & k wont make any difference insufficeint

from 1 & 2
yes sufficient to say that a=3,k=1& b=6 ,m=2 sufficient to say that a^k a factor of b^m
IMO C
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Re: If a, b, k, and m are positive integers, is a^k a factor of b^m?   [#permalink] 13 Jun 2019, 10:07
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