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

[victorgsiu]

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

Understand why A,D,B, are incorrect.

Is there a faster way to test C other than number plugging? These problem types take me about 2m 30sec unfortunately. Thanks.

[Ben Ku]

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

In order for a^k to be a factor of b^m, a^k must be divisible by b^m. That means that b^m needs to have k factors of a. For example, if a^k is 2^4, that means b^m must have four 2's.

The way I think of it is, can I simplify (b^m)/(a^k)? If I were to write this out, it would be:

(b*b*b* ...) / (a*a*a* ...)

First, the only way this would simplify is if a is a factor of b. Otherwise, we cannot cancel anything out. Secondly, if a is a unique factor of b, then we cannot have more a's than b's. So that means k must be less than or equal to m. We'll need both statements to be sufficient.

This is the more theoretical approach;I hope that it makes sense.