<p>"How many integers from 1 to 1000 inclusive are multiples of 7 and are equal to 5 times an even integer?"</p>
<p>I have no idea how to do those questions where I have to find the number of multiples of a number between two other numbers.</p>
<p>Help please</p>
<p>You have to realize that 5 times an even integer always ends in 0. So, count the number of multiples of 7 less than 1000 that end in 0. </p>
<p>Ideally, you’d have a graphing calculator, so you can use the table function with y=7x and just count the number of multiples that end in 0.</p>
<p>EDIT: There’s actually a way to solve this problem without a calculator, but I’ll leave that solution to other posters. Either way, realizing that “5 times an even integer” ends in 0 is crucial to solving this question.</p>
<p>You can write a number that fits the conditions as (7 * 5 * 2)n, or 70n. You know it’s a multiple of 70 because even if the even number is something else, like 4 or 6, it would still be a multiple of 70 ((7 * 5 * 4) = 140; (7 * 5 * 6) = 210). </p>
<p>As you can see, you’re looking for multiples of 70 that are between 1 and 1000. You can either count them out (which wouldn’t take very long), or set up a simple inequality:</p>
<p>1 <= 70n <= 1000.</p>
<p>1/70 <= n <= 14.29</p>
<p>n is all the numbers 1 through 14.</p>
<p>14 - 1 + 1 = 14 numbers that fit the condition.</p>
<p>So, 14.</p>
<p>And to answer your question, solving these problems takes creativity and keen observation. As pi noted, you can observe that each multiple ends in 0. Perhaps more helpfully, you can see that each number must be a multiple of 70. And that will lead you to your solution.</p>
<p>Also, practice will help you immensely. I studied number theory from AoPS which had increased my proficiency with these types of problems tremendously.</p>