# Divison In Mental ArithMatic

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 $687$?

• Divide 99548 by 687, call the remainder $r$. Then the number you want is $99548-r$ or $99548+687-r$, whichever one is nearer to 99548. – Gerry Myerson Aug 10 '16 at 13:08
• Do you have a reason to believe that there is a shorcut ? – Yves Daoust Aug 10 '16 at 13:13

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 $99615$.

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 $99548+67=99615$

This is not fast, but it's the way I do long division and it's guess-free. The point is, it isn't that hard to make a list of the first nine multiples of $$687$$

First set up the table.

$$\begin{array}{r|r} 1 & 687\\ 2 & \\ 3 & \\ 4 & \\ 5 & \\ 6 & \\ 7 & \\ 8 & \\ 9 & \end{array}$$

Multiples of $$2, 4, 8$$ can be found by successive doubling

$$\begin{array}{r|r} 1 & 687\\ 2 & 1374\\ 3 & \\ 4 & 2748\\ 5 & \\ 6 & \\ 7 & \\ 8 & 5496\\ 9 & \end{array}$$

Multiplying by $$5$$ can be accomplished by appending a $$0$$ to $$687$$ and dividing by $$2$$. $$6870 \div 2 = 3435$$

$$\begin{array}{r|r} 1 & 687\\ 2 & 1374\\ 3 & \\ 4 & 2748\\ 5 & 3435\\ 6 & \\ 7 & \\ 8 & 5496\\ 9 & \end{array}$$

$$3 \times 687$$ can be found by adding to two previous products. $$687 + 1374 = 2061$$.

$$6 \times 687$$ can be found by doubling $$3 \times 687$$.

$$\begin{array}{r|r} 1 & 687\\ 2 & 1374\\ 3 & 2061\\ 4 & 2748\\ 5 & 3435\\ 6 & 4122\\ 7 & \\ 8 & 5496\\ 9 & \end{array}$$

$$7 \times 687$$ can be found by adding $$687$$ to $$6 \times 687$$.

$$9 \times 687$$ can be found by adding $$687$$ to $$8 \times 687$$.

$$\begin{array}{r|r} 1 & 687\\ 2 & 1374\\ 3 & 2061\\ 4 & 2748\\ 5 & 3435\\ 6 & 4122\\ 7 & 4809\\ 8 & 5496\\ 9 & 6183 \end{array}$$

It takes me about a minute to do this. Now long division can proceed pretty fast.

## ALSO

Serendipity is also helpful at times.

Note that $$3 \times 687$$ is $$2061$$. Dividing $$99548$$ by $$2061$$ can be done quite quickly.

Ths answer is $$48$$ times with a remainder of $$620$$.

So $$99548 \div 687$$ is $$3 \times 48 = 144$$ with a remainder of $$620$$.

Since $$620$$ is closer to $$687$$ than it is to $$0$$, the nearest multiple of $$687$$ to $$99548$$ is $$99548 + (687 - 620) = 99615$$.