Write one digit on each side of the number 10 to create a four-digit multiple of 72. How many different solutions are there for this problem?

To solve this problem, we need to find a number that is a multiple of 72 and has the form _10_, where the underscores represent the digits we need to find. Since 72 is a multiple of both 8 and 9, any multiple of 72 must be divisible by both 8 and 9 as well.

Let's first consider the divisibility rule for 9: - A number is divisible by 9 if the sum of its digits is a multiple of 9.

To make _10_ divisible by 9, the sum of the digits including the two unknowns and the digits in the number 10 must also be a multiple of 9. This means that the sum of our four digits must be 9, 18, 27, and so on. Since 1 + 0 = 1, the unknown digits must add up to 8, 17, 26, etc.

Next, let's consider the divisibility rule for 8: - A number is divisible by 8 if its last three digits (in this case, the digits _10) form a number that is divisible by 8.

This means that for the number _10_, the digit in the tens place and the digit we add to the hundreds place must result in a number that is a multiple of 8.

Since a four-digit number can range from 1000 to 9999, and our number has the form _10_, the possible hundreds digits with the tens digit "1" can give us numbers ranging from x100 to x190, where x represents the hundreds place digit. We need to find which of those are divisible by 8.

We start by finding the multiples of 8 within that range. The closest multiple of 8 to 100 is 104, then 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, and 192. However, 192 is not valid as it exceeds 190. This leaves us with the multiples:

104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184

Now we must check which of these will also satisfy the divisibility rule for 9, meaning the sum of all the digits must be a multiple of 9.

Let's go through these candidates one by one:

- 1(104)0 = 1+0+4+0 = 5 (This is not a multiple of 9) - 1(112)0 = 1+1+2+0 = 4 (This is not a multiple of 9) - 1(120)0 = 1+2+0+0 = 3 (This is not a multiple of 9) - 1(128)0 = 1+2+8+0 = 11 (This is not a multiple of 9) - 1(136)0 = 1+3+6+0 = 10 (This is not a multiple of 9) - 1(144)0 = 1+4+4+0 = 9 (This is a multiple of 9) - 1(152)0 = 1+5+2+0 = 8 (This is not a multiple of 9) - 1(160)0 = 1+6+0+0 = 7 (This is not a multiple of 9) - 1(168)0 = 1+6+8+0 = 15 (This is not a multiple of 9) - 1(176)0 = 1+7+6+0 = 14 (This is not a multiple of 9) - 1(184)0 = 1+8+4+0 = 13 (This is not a multiple of 9)

From this list, only 1440 satisfies both divisibility rules for 8 and 9. Therefore, there is only one unique solution that adheres to the criteria of the problem, and the number is \( 11440 \).