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

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Joined: 06 Apr 2010
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If a, b, k, and m are positive integers, is a^k factor of [#permalink]

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23 Aug 2010, 07:52
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If a, b, k, and m are positive integers, is a^k factor of b^m?

(1) a is a factor of b.
(2) k = m
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Joined: 11 Aug 2010
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Re: If a, b, k, and m are positive integers, is a^k factor of [#permalink]

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23 Aug 2010, 08:46
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udaymathapati wrote:
If a, b, k, and m are positive integers, is a^k factor of b^m?
(1) a is a factor of b.
(2) k = m

Ans C

1) if a is factor of b then b = a x N (where N is factor of B other than a )
well this is not sufficient since we dont know the value of k and m .

2 ) k=m alone not sufficient to decide if a^k is a factor of b^m or not.

Together we can say a is a factor of b and if k=m then b^m have a^m (which is equal to a^k) as its factor !
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Re: If a, b, k, and m are positive integers, is a^k factor of [#permalink]

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23 Aug 2010, 08:58
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udaymathapati wrote:
If a, b, k, and m are positive integers, is a^k factor of b^m?
(1) a is a factor of b.
(2) k = m

Question: is $$\frac{b^m}{a^k}=integer$$?

(1) a is a factor of b --> $$ax=b$$, where $$x$$ is a positive integer --> is $$\frac{a^m*x^m}{a^k}=a^{m-k}*x^m=integer$$, now if $$m-k\geq{0}$$ then $$a^{m-k}*x^m$$ will be an integer BUT if $$m-k<{0}$$ then $$a^{m-k}*x^m$$ may NOT be an integer (well it will be an integer if $$a=1$$, but if $$a\neq{1}$$, then no). Not sufficient.

(2) k = m --> is $$\frac{b^k}{a^k}=(\frac{b}{a})^k=integer$$? --> if $$\frac{b}{a}=integer$$ then the answer would be YES, but if $$\frac{b}{a}\neq{integer}$$ then the answer would be NO. Not sufficient.

(1)+(2) $$ax=b$$ and $$k=m$$ --> $$\frac{b^m}{a^k}=\frac{a^k*x^k}{a^k}=x^k=integer$$. Sufficient.

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

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14 Aug 2017, 05:24
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Re: If a, b, k, and m are positive integers, is a^k factor of [#permalink]

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24 Aug 2017, 12:39
udaymathapati wrote:
If a, b, k, and m are positive integers, is a^k factor of b^m?

(1) a is a factor of b.
(2) k = m

We are given that a, b, k, and m are positive integers, and we need to determine whether a^k is a factor of b^m.

Statement One Alone:

a is a factor of b.

Although we know that a is a factor of b, we still cannot determine whether a^k is a factor of b^m.

For example, if a = 2, b = 4 (2 is factor of 4), k = 1, and m = 1, then 2^1 = 2 is a factor of 4^1 = 4.

However, if a = 2, b = 4, k = 3, and m = 1, then 2^3 = 8 is not factor of 4^1 = 4. Statement one alone is not sufficient.

Statement Two Alone:

k = m

Since we know neither the values of a and b nor the relationship between a and b, statement two alone is not sufficient.

Statements One and Two Together:

From statement one, we know a is a factor of b, and from statement two, we know k = m.

In order for a^k to be a factor of b^m, (b^m)/(a^k) = integer.

Since k = m, we can write (b^m)/(a^k) as (b^m)/(a^m). Now let’s simplify:

(b^m)/(a^m) = (b/a)^m

Since a is a factor of b, b/a is an integer; thus:

(b/a)^m = integer^m = integer

Thus, a^k is indeed a factor of b^m and the two statements together are sufficient.

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Re: If a, b, k, and m are positive integers, is a^k factor of   [#permalink] 24 Aug 2017, 12:39
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