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# Number Theory

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When General Han counts the soldiers in his army, he uses the following method. He orders them to line up in rows of 9, then in row of 10, and finally, in rows of 11, and each time he counts the number of soldiers not in a row. One morning, he finds that there are 7 soldiers left when the rest are in groups of 9, 5 soldiers left when the rest are in rows of 10, and 5 soldiers left when the rest are in rows of 11. He knows that there are 1000 soldiers in his army. How many of the soldiers are present this morning?

Feb 4, 2022

#1
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There are 248 soldiers.

Feb 4, 2022
#2
+2437
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$$n\leq 1000$$

You have...

$$7 \pmod{9}$$

$$5 \pmod {10}$$

$$9 \pmod {11}$$

Plug into a calculator, and you find that there were $$\color{brown}\boxed{295}$$

Feb 4, 2022
#3
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N mod 9==7

N mod 10==5

N mod 11==5

By Chinese Remainder Theorem, you get:

N ==115 soldiers.

Guest Feb 4, 2022