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Abstract Algebra

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Chinese Remainder Theorem Word Problem

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Fred has bags of three different sizes-- small, medium, and large, from which he sells apples. The small size can hold 3 apples, the medium size can hold 7, and the large size can hold 8.

Fred picked apples from an orchard with 167 apples. When he tried to fit all his apples in small bags, he had one left over. When he tried to fit all his apples in medium bags, he had 4 left over. When he tried to fit all his apples in large bags, he had 3 left over.

How many apples did Fred pick?

A

$66$

B

$67$

C

$134$

D

$135$