Problem 21 · 2024 Math Kangaroo
Stretch
Number Theory
divisibilityfactor-pairs
How many integers k have the property that \(k+6\) is a multiple of \(k-6\)?
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Answer: E — 12
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Hint 1 of 2
Write k + 6 as (k - 6) + 12.
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Hint 2 of 2
Then k - 6 must divide 12; count all its divisors, positive and negative.
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Approach: reduce to (k-6) divides 12 and count divisors
- Since k + 6 = (k - 6) + 12, the condition is that (k - 6) divides 12.
- The divisors of 12 are +-1, +-2, +-3, +-4, +-6, +-12, which is 12 values.
- Each gives one integer k, so there are 12 such integers.
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