Problem 15 · 2016 AMC 8
Medium
Number Theory
difference-of-squaresfactorization
What is the largest power of 2 that is a divisor of 134 − 114?
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Answer: C — 32.
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Hint 1 of 3
Resist computing 134 − 114 (that's 28561 − 14641 — a big number you'd then have to factor). Counting factors of 2 only needs the number in FACTORED form, so factor it BEFORE you multiply anything out.
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Hint 2 of 3
It's a difference of two fourth powers, which is a difference of squares: a4 − b4 = (a2 + b2)(a2 − b2). That breaks it into small numbers you can factor by eye.
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Hint 3 of 3
Then just tally the 2s: the largest power of 2 dividing a product is found by adding up the factors of 2 in each piece.
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Approach: factor first (difference of squares), then count factors of 2
- 134 − 114 = (132 + 112)(132 − 112) = (169 + 121)(169 − 121) = 290 · 48 — two small numbers instead of one huge one.
- Count 2s in each: 290 = 2 · 145 contributes one factor of 2; 48 = 24 · 3 contributes four.
- Add the exponents: 21+4 = 25 = 32.
- Why this transfers: to find the highest power of a prime dividing a product, never multiply it out — factor each piece and ADD the exponents of that prime. Difference-of-squares (and its repeat) is the standard tool for cracking an − bn open.
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