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?
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
