1980 Canadian MO Problems/Problem 1

Problem

If $a679b$ is the decimal expansion of a number in base $10$ , such that it is divisible by $72$ , determine $a,b$ .

Solution

Write $a679b$ as $a679b = 10000a + 6790 + b = 72n$ , for some integer $n$ . Since $72 = 8 \times 9$ , we look at this equation $\mod 8$ , then $\mod 9$ .

Looking at the equation $\mod 8$ :

$\begin{matrix} 10000a + 6790 + b \equiv 0 + 6 + b &\equiv& 0 \mod 8 \\ 6 + b &\equiv& 0 \mod 8 \\ b &\equiv& 2 \mod 8 \end{matrix}$

Thus, $b = 2$ .

Substituting $b=2$ into the equation, and looking at the equation $\mod 9$ :

$\begin{matrix} 10000a + 6790 + b \equiv a + 6792 &\equiv& 0 \mod 9 \\ a+6 &\equiv& 0 \mod 9 \\ a &\equiv& 3 \mod 9 \end{matrix}$

Thus, $a = 3$ .

Thus, we get the 5 digit number $36792$ , which is indeed divisible by $72$ , thus satisfying requirements. Thus, $a = 3, b = 2$ .

1980 Canadian MO (Problems)
Preceded by First Question	1 • 2 • 3 • 4 • 5	Followed by Problem 2

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1980 Canadian MO Problems/Problem 1

Problem

Solution

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