Question

Write one digit on each side of 10 to make a four digit multiple of 72. How many different solutions does this problem have?

Answer 1

• 4104
• one solution

Explanation:

72 is the product of 8 and 9, so the number needs to be divisible by both 8 and 9.

The number 100 is divisible by 8 with a remainder of 4, so to make it have no remainder, we must add another 4 to the number: 104. We cannot have any other digit in the ones place (x) and have 10x be divisible by 8.

The 4-digit number must have a sum of digits that is divisible by 9 if it is to be divisible by 9. 104 has a sum of digits of 5, so we must have another digit of 4 to get a number divisible by 9. There is no other single digit that will do.

The number we're looking for is 4104 = 72·57. There are no other solutions.