Write one digit on the both sides of 57 to make the number divisible by 72. How many solutions does this problem have?

Answer:   2 solutions: 0576, 9576Step-by-step explanation:For the number to be divisible by 72, it must be divisible by 9 and by 8. __divisible by 8The 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__divisible by 9The 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__solution valuesThe above suggests two solutions to the problem:   0576 = 8 × 72   9576 = 133 × 72