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
Here my explanation.
Stem 1:
a is a factor of b.
If a is a factor of b then i have found that there power will be factor of each other.
Suppose a=2 and b=4. here a is a factor of b.
now k=2 and m=2, then a=4 and b=16, hence a is a factor of b.
k=3 and m=4 then a=8 and b=256, hence a is a factor of b.
Sufficient.
Stem 2:
k is less then or equal m.
Here we can take different value for 'a' and 'b'
Insufficient.
Answer is A.
Where i have made a mistake? Is it correct?
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Data sufficiency
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GMATGuruNY
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Statement 1: a is a factor of bNewaz111 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
Test one case that also satisfies statement 2.
Case 1: a=2, b=2, k=1, m=1
In this case, a^k = 2 and b^m = 2.
Since a^k=2 is a factor of b^m=2, the answer to the question stem is YES.
Test one case that DOESN'T also satisfy statement 2.
Case 2: a=2, b=2, k=2, m=1
In this case, a^k = 4 and b^m = 2.
Since a^k=4 is NOT a factor of b^m=2, the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.
Statement 2: k ≤ m
Case 1 also satisfies statement 2.
In Case 1, the answer to the question stem is YES.
Case 3: a=2, b=1, k=1, m=1
In this case, a^k = 2 and b^m = 1.
Since a^k=2 is NOT a factor of b^m=1, the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 3, INSUFFICIENT.
Statements combined:
Statement 1 implies that b is a multiple of a:
Let b = ax, where x is a positive integer.
Statement 2 implies that m ≥ k.
Let m = k + y, where y is a nonnegative integer.
Thus:
b^m = (ax)^(k+y) = (ax)^k * (ax)^y = (a^k)(x^k)(a^y)(x^y).
The value in red indicates that a^k is a factor of b^m.
SUFFICIENT.
The correct answer is C.
Last edited by GMATGuruNY on Fri Jun 05, 2015 10:22 am, edited 1 time in total.
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Hi Newaz111,
Your instinct to TEST VALUES here is a good one. When TESTing VALUES for multiple variables though, you have to be careful about any 'unconscious' patterns that you apply to those variables.
In Fact 1, we're told that A is a factor of B, so A MUST be less than or equal to B. The other two variables (K and M) can be ANY positive integers though...In one of your examples, you made the two variables EQUAL, in another, you made M slightly bigger than K.....but what other options are there.....
Couldn't K be MUCH BIGGER than M? And if that were to occur, what would the answer to the question be? As you continue to practice, you'll hone these skills - just continue to be on the lookout for the various options that can occur with your TESTS.
GMAT assassins aren't born, they're made,
Rich
Your instinct to TEST VALUES here is a good one. When TESTing VALUES for multiple variables though, you have to be careful about any 'unconscious' patterns that you apply to those variables.
In Fact 1, we're told that A is a factor of B, so A MUST be less than or equal to B. The other two variables (K and M) can be ANY positive integers though...In one of your examples, you made the two variables EQUAL, in another, you made M slightly bigger than K.....but what other options are there.....
Couldn't K be MUCH BIGGER than M? And if that were to occur, what would the answer to the question be? As you continue to practice, you'll hone these skills - just continue to be on the lookout for the various options that can occur with your TESTS.
GMAT assassins aren't born, they're made,
Rich
