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This problem trains for: AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8.

How many positive integers give the same quotient as remainder when divided by 44?

Apply the theorem of integer division to find the quotient q and the remainder r:

equation

Since q=r we can use factoring to write:

equation

The numbers we are looking for are multiples of 45. However, we must remember that the theorem of integer division asks for the remainder to be smaller than the divisor. Therefore, only remainders from 0 to 43 are possible:

equation

There are 44 multiples of 45 with this property.