Answer:
2 solutions: 0576, 9576
Explanation:
For the number to be divisible by 72, it must be divisible by 9 and by 8.
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divisible by 8
The number will be divisible by 8 if the last 3 digits are divisible by 8. That is, for digit n appended on the right, we require ...
(570 +n) mod 8 = 0
570 mod 8 + n mod 8 = 0
2 + n mod 8 = 0
n = -2 mod 8 = 6 . . . . one solution
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divisible by 9
The number will be divisible by 9 if the sum of all of the digits is divisible by 9. Where n is the digit appended on the left, we require ...
(n + 5 + 7 + 6) mod 9 = 0
n mod 9 = 0
n = 0 or 9 . . . . two solutions
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solution values
The above suggests two solutions to the problem:
0576 = 8 × 72
9576 = 133 × 72