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Question

A Chinese emperor orders a regiment of soldiers in his palace to divide into groups of 4. They do so successfully. He then orders them to divide into groups of 3, upon which 2 of them are left without a group. He then orders them to divide into groups of 11, upon which 5 are left without a group. If the emperor estimates there are about two hundred soldiers in the regiment, what is the most likely number of soldiers in the regiment?

Accepted Answer

Let n be the number of soldiers. According to the problem statement, it follows that align* n & 0 4 n & 2 3 n & 5 11 align*By the Chinese Remainder Theorem, there is an unique residue that n can leave, modulo 33; since 5 2 3, it follows that n 5 33. Also, we know that n is divisible by 4, so by the Chinese Remainder Theorem again, it follows that n 104 132. Writing out the first few positive values of n, we obtain that n = 104, 236, 368, and so forth. The closest value of n is 236.