- When the number $357a$ is divided by $5$, the remainder is $2$
Evaluate the values $a$ can take.
I'm aware of the fact that we could just take a look into divisibility rule of $5$, which yields $a \in \{0,5\}$. Since the remainder is $2$, we get that $a \in \{2,7\}$. What about using modular arithmetics?
$$3000 + 500 + 70 + a \equiv 2 \pmod {5}$$
After simplification
$$ a \equiv 2 \pmod {5}$$
I don't know whether it is correct. Can you assist?
Regards
Just to confirm, your answer is completely correct, and must have resulted from the fact that $3000,500$ and $70$ are divisible by $5$, so these terms vanish from the left hand side, leaving $a \equiv 2 \mod 5$.
Another equivalent way of solving this problem, is that instead of breaking into powers of $10$ (digit expansion), you simply note that $357a - a = 3570 = 5 \times (2 \times 357)$ is a multiple of $5$, so by the definition of congruence, $357a \equiv a \mod 5$ directly follows , so $a \equiv 2 \mod 5$, and $a$ is a single digit number so we can conclude.