Divison In Mental ArithMatic

Question

The nearest number to $99548$ which is divisible by $687$ is?

How can I find the answer quickly, is there any short cut to check if a number is divisible by $687$?

Answer 1

It isn't all that hard mentally. As Ross Millikan says, $687=3\cdot 229$, so what you are looking for is the multiple of $229$ which is nearest to $33182\frac{2}{3}$.

The key to happiness is that $230$ is close to $229$, and it only has two digits.

Divide $3318.2$ by $23$, which gives you $143$. Multiply back by $230$, giving $32890$, and subtract $143$ to get $32747$, a multiple of $229$. Subtracting, you are $33182-32747=435$ short of $33182$, so add $229$ twice to get $33205$.

Before you feel too pleased with yourself, remember to multiply back by $3$, to get $99615$.

Answer 2

As $687=3\cdot 229$ and $229$ is a rather large prime, you aren't going to find a nice divisibility test. Even if you had one, you wouldn't just want to try numbers near $99548$ until you find one. The best I can suggest is the usual division with remainder. That will be tough mentally, depending on how much practice you have and how many numbers you can keep track of, but pencil and paper will be pretty easy. In this case $99548 \equiv 620 \pmod {687}$, so you want $99548+67=99615$