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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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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?
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13 Jun 2019, 04:27
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)) Answer: C.
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Re: If a, b, k, and m are positive integers, is a^k a factor of b^m?
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13 Jun 2019, 10:07
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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If a, b, k, and m are positive integers, is a^k a factor of b^m?
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08 Sep 2019, 18:48
Hi Bunuel GMATPrepNow VeritasKarishma EMPOWERgmatRichCIs my below understanding correct w.r.t. Bunuel 's solution above: 1. An integer raised to a positive integer MUST to be an integer. 2. An integer raised to a negative integer may / may not be an integer. e.g. 2^ (1) is not an integer, whereas 1^ (1) is an integer. This is where St 2 really plays an important role. Am I correct? Can I not play with picking nos in such problems with given restrictions in qs stem since I can not choose negative no and 0?
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Re: If a, b, k, and m are positive integers, is a^k a factor of b^m?
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08 Sep 2019, 20:54
Hi adkikani, TESTing VALUES works on lots of different Quant questions (in both PS and DS) and it can certainly be used on this prompt. After doing a little work, you might actually discover a Number Property in this prompt (although you don't need to use that rule to get the correct answer). We're told that A, B, K and M are all POSITIVE INTEGERS. We're asked if A^K is a factor of B^M. This is a YES/NO question. 1) A is a factor of B. IF.... A=B=2 and K=M=1.... 2^1 = 2 is a factor if 2^1 = 2 and the answer to the question is YES A=B=2 and K=2, M=1.... 2^2 = 4 is NOT a factor if 2^1 =2 and the answer to the question is NO Fact 1 is INSUFFICIENT 2) K ≤ M. IF.... A=B=2 and K=M=1.... 2^1 = 2 is a factor if 2^1 = 2 and the answer to the question is YES A=3, B=2 and K=M=1.... 3^1 = 3 is NOT a factor if 2^1 = 2 and the answer to the question is NO Fact 2 is INSUFFICIENT Combined, we know... A is a factor of B. K ≤ M Since A is a factor of B, we know that A divides evenly into B. In that situation, the only way for A^K to NOT be a factor of B^M would be when K is greater than M (you may have already noticed that in the work that we did above). Since K must be less than/equal to M, A^K will always be a factor of B^M and the answer to the question is ALWAYS YES. Combined, SUFFICIENT. Final Answer: GMAT assassins aren't born, they're made, Rich
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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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Updated on: 09 Sep 2019, 03:27
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. Given: a, b, k, and m are positive integers Asked: Is a^k a factor of b^m? (1) a is a factor of b. Even if a is a factor of b, If k>m, then a^k may or may not be a factor of b^m NOT SUFFICIENT (2) k ≤ m. Since relation between a and b is unknown NOT SUFFICIENT Combining (1) & (2) (1) a is a factor of b. (2) k ≤ m. a^k will be a factor of b^m since b = a n \(b^m = a^m * n^m = a^k a^{mk} * n^m\) a^k is factor of b^m SUFFICIENT IMO C
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Originally posted by Kinshook on 08 Sep 2019, 22:17.
Last edited by Kinshook on 09 Sep 2019, 03:27, edited 2 times in total.



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Re: If a, b, k, and m are positive integers, is a^k a factor of b^m?
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09 Sep 2019, 03:24
adkikani wrote: Hi Bunuel GMATPrepNow VeritasKarishma EMPOWERgmatRichCIs my below understanding correct w.r.t. Bunuel 's solution above: 1. An integer raised to a positive integer MUST to be an integer. 2. An integer raised to a negative integer may / may not be an integer. e.g. 2^ (1) is not an integer, whereas 1^ (1) is an integer. This is where St 2 really plays an important role. Am I correct? Can I not play with picking nos in such problems with given restrictions in qs stem since I can not choose negative no and 0? Yes, an integer raised to a positive integer will be an integer. Since a positive integer exponent is just the base multiplied by itself as many times as specified in the power, \(4^5 = 4*4*4*4*4\)  it will remain an integer. An integer raised to a negative power may or may not be an integer. Correct as shown by you in your examples. I did the question by thinking of number plugging only here. It seems easy that way. If a, b, k, and m are positive integers, is a^k a factor of b^m? Is \(\frac{b^m}{a^k} = \frac{b*b*b*b* ...}{a*a* ...}\) an integer? This will be an integer when all a's get cancelled by the b's above i.e. if it were something like this \(\frac{15*15*15*15}{5*5*5}\). (1) a is a factor of b. This is necessary but not sufficient. 5 is a factor of 15 in example above. But what if we have four 15s but six 5s? Then all 5s will not cancel out. (2) k ≤ m. This helps but is not sufficient. What if k is 3 as shown in the example above but instead of 5s in denominator, we have 100s? Then we will not get an integer. Using both, a is a factor of b and we have fewer a's. So it will be something similar to the fraction above and we will get an integer. Answer (C)
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Re: If a, b, k, and m are positive integers, is a^k a factor of b^m?
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