Q13/007) Let P = 2 * 4 * 6 * ... * 400 Let Q = 1 * 3 * 5 * ....* 399 Prove that (P-Q) is divisible by 401.

Solution

we know 

2 = - 399 mod 401

4= - 397 mod 401

6 = - 395 mod  401

.

.

400 = -1 mod 401

multiply all and rearrange all terms ( knowing that there are 200 - signs) we get the result