05/30/2020 (2011: National Team Round, Problem 5)

Q: How many three-digit integers are divisible by the digit in their ones place?

A: 420

All integers (besides 0) are divisible by 1, so all integers with a ones digit of 1 are divisible by their ones digits. There are 9 options for the hundreds digit and 10 options for the tens digit, making there be 9 * 10 3 digit integers with a units digit of 1, which is 90 integers.

All integers with a units digit of 2 are divisible by 2, their ones digit.  There are 9 options for the hundreds digit and 10 options for the tens digit, making there be 9 * 10 3 digit integers with a units digit of 2, which is 90 integers.

NOT all integers with a units digit of 3 are divisible by 3, so now we have to actually use our brains. There are 3 ways to make the hundreds and tens digits add up to 3 (3,0; 2,1;1,2). There are 6 ways to make the hundreds and tens digits add up to 6 (6,0; 5,1; 4,2; 3,3; 2,4; 1,5). There are 9 ways to make the hundreds and tens digits add up to 9 (9,0; 8,1; 7,2; 6,3; 5,4; 4,5; 3,6; 2,7; 1,8). There are 7 ways to make the hundreds and tens digits add up to 12 (9,3; 8,4; 7,5; 6,6; 5,7; 4,8; 3,9). There are 4 ways to make the hundreds and tens digits add up to 15 (9,6; 8,7; 7,8; 6,9). There is one way ro make the hundreds and tens digits add up to 18 (9,9). You add up 3 + 6 + 9 + 7 + 4 + 1 = 30. There are 30 integers.

NOT all integers with a units digit of 4 are divisible by 4, so now we have to use our brains again. There are 9 options for the hundreds digit, but only 5 options for the tens digit (0;2;4;6;8). That means there are 9 * 5 = 45 integers.

All integers with a units digit of 5 are divisible by 5, their ones digit.  There are 9 options for the hundreds digit and 10 options for the tens digit, making there be 9 * 10 3 digit integers with a units digit of 5, which is 90 integers.

NOT all integers with a units digit of 6 are divisible by 6, so now we have to use our brains again (this seems to be a recurring theme). You have to make sure that the the hundreds and tens digits add up to a multiple of 3. There are 3 ways to make the hundreds and tens digits add up to 3 (3,0; 2,1;1,2). There are 6 ways to make the hundreds and tens digits add up to 6 (6,0; 5,1; 4,2; 3,3; 2,4; 1,5). There are 9 ways to make the hundreds and tens digits add up to 9 (9,0; 8,1; 7,2; 6,3; 5,4; 4,5; 3,6; 2,7; 1,8). There are 7 ways to make the hundreds and tens digits add up to 12 (9,3; 8,4; 7,5; 6,6; 5,7; 4,8; 3,9). There are 4 ways to make the hundreds and tens digits add up to 15 (9,6; 8,7; 7,8; 6,9). There is one way ro make the hundreds and tens digits add up to 18 (9,9). You add up 3 + 6 + 9 + 7 + 4 + 1 = 30. There are 30 integers.

NOT all integers with a units digit of 7 are divisible by 7, and there isn't an 'easy' rule to use. The rule to see if a number is divisible by 7 is to double the units digit (7) and subtract it from the rest of the number, which will end up a multiple of 7 if it is divisible by 7. As of this, the least possible 3 digit multiple of 7 with a units digit of 7 is 147, and you add 70s until you reach the greatest 3 digit multiple of 7 with a units digit of 7, which is 987. That means that there are 13 numbers with a units digit of 7 that are multiples of 7.

NOT all integers with a units digit of 8 are divisible by 8. The lowest multiple of 8 with a units digit of 8 is 128 and you keep on adding 40s until you reach the greatest 3 digit number you can get, which is 968. That means there are (968-(128-40))/40 numbers, which is equal to 22 integers.

Lastly, we have integers with the units digit of 9 (we aren't doing numbers with a units digit of 0 because NO number is divisible by 0). The numbers have to add up to a multiple of 9. We start with 18, because the only way to have a units digit of 9 and to have the numbers add up to 9 is to have 009, which is not 3 digits. There are 9 ways to make the digits add up to 18 (189, 279, 369, 459, 549, 639, 729, 819, 909). There is one way to make the digits add up to 27 (999). 9 + 1 = 10 integers.

You add up all of them together and get: 90 + 90 + 30 + 45 + 90 + 30 + 13 + 22 + 10 = 420 integers