Find the number of four-digit positive integers divisible by or .

Source: NCTM Mathematics Teacher, August 2006

**Solution**

Consider , the set of four-digit positive integers divisible by and , the set of four-digit positive integers divisible by . The count of and the count of .

Some integers like and appear in both sets so we subtract them as duplicates from the total count. Since , starting from the duplicates occur every th integer in set or every rd integer in set . If we use set , the count of duplicates = . The number of four-digit positive integers divisible by and/or is .

**Answer**:

**Alternative solution**

If we divide by , we get which means that there are integers that are divisible by . But, if we divide by , we get which is not an integer. So we say there are integers that are divisible by . To avoid this problem we use the floor function = the largest integer less than or equal to for some real number . In our example, .

Number of four-digit integers divisible by

Number of four-digit integers divisible by

Number of four-digit integers divisible by

There are four-digit integers divisible by and/or .