Question

how many four-digit integers either leave a remainder of 2 when divided by 7, or a remainder of 4 when divided by 5, but not both

Answer 1

By the Inclusion-Excluson Principle, the number of positive integers that

satisfy x≡2(mod7)

or x≡4(mod5)

is 1286+1800−257=2829

The number that satisfy exactly one of these congruences is 1286+1800−2⋅257=2572