2005 AMC 12B Problems/Problem 21
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Problem
A positive integer has divisors and has divisors. What is the greatest integer such that divides ?
Solution
If has factors, then is a product of powers of (not necessarily distinct) primes. When multiplied by , the amount of factors of increased by , so there are possible powers of in the factorization of , and possible powers of in the factorization of , which would be , , and . Therefore the highest power of that could divide is .
See also
2005 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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