Answer
a. Yes
b. No
Work Step by Step
a. Yes, since 7 integers > 5 possible remainders. There are 7 pigeons for 5 pigeonholes, so at least 2 will have the same remainder.
Remainders when divided by 6 = {0, 1, 2, 3, 4, 5}
1 mod 6 = 1
2 mod 6 = 2
3 mod 6 = 3
4 mod 6 = 4
5 mod 6 = 5
6 mod 6 = 0
7 mod 6 = 1
8 mod 6 = 2
9 mod 6 = 3
b. No, since 7 integers < 8 possible remainders. There are less pigeons than pigeonholes.
Remainders when divided by 8 = {0, 1, 2, 3, 4, 5, 6, 7}
