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What is the largest number $k$ less than 1000 such that the remainder is 1 when $k$ is divided by each of 3, 4, 5, 6, and 7?

 Dec 23, 2020
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Using Chinese Remainder Theorem + Modular Multiplicative Inverse, we have:

 

K mod 3 = 1,   

K mod 4 = 1,   

K mod 5 = 1,   

K mod 6 = 1,   

K mod 7 = 1

 

K =420m  +  1, where m=0, 1, 2, 3........etc.

 

So, the smallest K under 1000=420 + 1 =421

 Dec 23, 2020

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