2022 AMC 10A Problems/Problem 19
Problem
Define as the least common multiple of all the integers from to inclusive. There is a unique integer such that
What is the remainder when is divided by ?
Solution
Notice that contains the highest power of every prime below . Thus, .
When writing the sum under a common fraction, we multiply the denominators by divided by each denominator. However, since is a multiple of , all terms will be a multiple of until we divide out , and the only term that will do this is . Thus, the remainder of all other terms when divided by will be , so the problem is essentially asking us what the remainder of divided by is. This is equivalent to finding the remainder of divided by .
We use modular arithmetic to simplify our answer:
This is congruent to
Evaluating, we get:
Therefore the remainder is and the answer is
~KingRavi
Video Solution By ThePuzzlr
~ MathIsChess
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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