Question

How many three digit counting numbers are not multiples of 10?

Answer 1

Three digits number go from 100 to 999, and thus there are 999-100+1 = 900 of them.

The multiples of ten are numbers ending with zero, so we must exclude all numbers like xy0.

The first digit, x, range from 1 to 9, while the second digit, y, range from 0 to 9.

So, to generate a multiple of ten between 100 and 999, we have 9 choices for the first digit, 10 choices for the second, while the third digit is fixed to zero.

This means that there are $9 \cdot 10 = 90$ multiples of ten between 100 and 999.

Since there are 900 numbers in total, there are 900-90 = 810 numbers between 100 and 999 that are not multiples of ten.