How many integers between \(100\) and \(2024\) are multiples of both \(5\) and \(7\), but are not multiples of \(10\)?
The integers that are multiples of both \(5\) and \(7\) are the integers that are multiples of \(35\). Now let’s determine which multiples of \(35\) are also multiples of \(10\). Notice that \(1 \times 35 = 35\), which is not a multiple of \(10\). However, \(2 \times 35 = 70\), which is a multiple of \(10\). In fact, multiplying \(35\) by any even integer will result in a multiple of \(10\). This is because \(35\) is a multiple of \(5\), and all even integers are multiples of \(2\). So multiplying \(35\) by an even integer will result in an integer which is a multiple of both \(5\) and \(2\), and thus a multiple of \(10\). So if we are looking for integers that are multiples of \(35\) but not multiples of \(10\), then we must multiply \(35\) by odd integers only.
The smallest multiple of \(35\) greater than \(100\) is \(3 \times 35 = 105\). Similarly, the largest multiple of \(35\) less than \(2024\) is \(57 \times 35 = 1995\). It follows that the number of integers between \(100\) and \(2024\) that are multiples of both \(5\) and \(7\), but are not multiples of \(10\), is equal to the number of odd integers between \(3\) and \(57\), inclusive. This is equal to the number of odd integers between \(1\) and \(55\), inclusive. We know that exactly half of the integers between \(1\) and \(54\) are odd, and \(55\) is an odd integer. So in total, there are \(\frac{54}{2}+1=27+1=28\) odd integers between \(1\) and \(55\), inclusive.
Thus, there are \(28\) integers
between \(100\) and \(2024\) that are multiples of both \(5\) and
\(7\), but are not multiples of \(10\).