How many two-digit counting numbers are not multiples of $5 ?$ There are two-digit counting numbers which are not multiples of 5 .

Step 1 ：The two-digit counting numbers range from 10 to 99. We need to find out how many of these are not multiples of 5.

Step 2 ：A number is a multiple of 5 if its last digit is either 0 or 5. So, we need to count the numbers which do not end with 0 or 5.

Step 3 ：After counting, we find that there are 72 such numbers.

Step 4 ：Final Answer: \(\boxed{72}\)