Multiple choice: $S = {x | 0 ≤ x < 280 ∧ x ≡ 3 (mod 7) ∧ x ≡ 4 (mod 8)}$ The question is:
Consider the following set of integers:
$$ S = \left\{x \left| 0 \le x < 280
                       ∧ x \equiv 3 \mod 7
                       ∧ x \equiv 4 \mod 8 \right. \right\}. $$
How many integers are there in S?
$0$? $1$? $2$? $5$? $10$? $280$? 


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*It's a multiple choice answer with the correct answer of 5. I just can't seem to wrap around the idea.


$x \equiv 3 \pmod{7}$ is equivalent to $x+4 \equiv 0 \pmod{7}$.
$x \equiv 4 \pmod{8}$ is equivalent to $x+4 \equiv 0 \pmod{8}$.
Thus we want to know which numbers $x$ are such that $x+4$ is divisible by both $7$ and $8$. Since $7$ and $8$ are coprime, it would be exactly those numbers such that $x+4$ is divisible by $7 \times 8 = 56$. You can count easily now.
