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The five-digit number $\overline{a679b}$ is divisible by 72.  Find the digits a and b.

 Feb 5, 2021
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Since the number is divisible by 72, it must be divisible  8 and also by 9. For it to be divisible by 9, sum of its digits must also do so; so a + 6 + 7 + 9 + b= a + b + 22 must be divisible by 9, that is, a+ b=5  or a+b = 14. For the number to be divisible by 8, its rightmost 3 digits must make a a number divisible by 8; so 79b must be divisible by 8. Since 79b = 8(90) + 7b, the number 7b must also be divisible by 8. The only possibility for b is 2, and a must, therefore, be 3. So a679b = 36792=72(511).

 Feb 6, 2021

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