Question

find the remainder when (17)(9!) + (2)(18!) is divided by (9!)17408​

Answer 1

9^17 can be written as (7+2)^17, binomial expansion of which leaves only one term (2^17) without 7.

So the expression (7+2)^17 mod 7 simplifies to 2^17 mod 7.

2^17 mod 7 = (2^3)^5 × 2^2 mod 7

ie 8^5 × 4 mod 7

ie (7+1)^5 × 4 mod 7

This simplifies to 1^5 × 4 mod 7

i.e. 4 mod 7 = 4