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

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

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New post 23 Aug 2010, 08:52
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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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Re: If a, b, k, and m are positive integers, is a^k factor of  [#permalink]

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New post 23 Aug 2010, 09:46
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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 !
Answer is C
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Re: If a, b, k, and m are positive integers, is a^k factor of  [#permalink]

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New post 23 Aug 2010, 09:58
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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.

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

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New post 24 Aug 2017, 13:39
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.

Answer: C
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Re: If a, b, k, and m are positive integers, is a^k factor of &nbs [#permalink] 24 Aug 2017, 13:39
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