How many integers have the same quotient and remainder after being divided by 13?

How many integers have the same quotient and remainder after being divided by 13?

Let the remainder be X after being divided by 13, then the number with the same quotient and remainder after being divided by 13 is: 13X + x = (13 + 1) x = 14x, because x < 13, so 100 < 14x < 182, the solution is 507 < x < 13, that is, x can be 8, 9, 10, 11, 12; so there are five numbers with the same quotient and remainder after being divided by 13, namely 13 × 8 + 8 = 112, 13 × 9 + 9 = 126, 13 × 10 + 10 = 140, 13 × 11 + 11 = 154, 13 × 12 + 12 = 168 There are five numbers with the same division quotient and remainder