H1. The five-digit integer ' $a 679 b$ ' is a multiple of 72 . What are the values of $a$ and $b$ ?
Solution
Since $72=8 \times 9$ any number that is a multiple of 72 is both a multiple of 8 and a multiple of 9 . Conversely, because 8 and 9 have no common factor greater than 1 and $8 \times 9=72$, any number that is both a multiple of 8 and a multiple of 9 is a multiple of 72 .
A number is a multiple of 8 if, and only if, its last three digits form a multiple of 8 . Hence ' $79 b^{\prime}$ ' is a multiple of 8 . Since 800 is a multiple of 8 , the only number of the required form is 792, and thus $b=2$.
A number is a multiple of 9 if, and only if, the sum of its digits is a multiple of 9 . Hence $a+6+7+9+b$ is a multiple of 9 . Because $b$ is 2 , it follows that $a+24$ is a multiple of 9 , and hence $a=3$.
Hence $a=3$ and $b=2$.