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The five digit number a986b is divisible by 72. What is the value of a+b ?
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72 = 9*8.
The sum of digits S = a + 9 + 8 + 6 + b must be multiple of 9 and the number 86b must be multiple of 8.
See the divisibility by 9 rule in the lesson Divisibility by 9 rule in this site.
The divisibility by 8 rule says that the number 86b is multiple of 8.
From this condition ("86b must be multiple of 8") we have with necessity that b = 4. There is no other possibility.
Now, S = 23 + (a + b) = 23 + 4 + a = 27 + a.
a >= 1 and <= 9.
Since S is multiple of 9, there is only one possibility for a:
a = 9.
Answer. a = 9, b= 4 and a + b = 9 + 4 = 13.
