Question

A certain number when it divides 62 leaves 2, when it divides 80 leaves 5. What is the greatest possible value of the number?

Answer 1

Here we essentially have the following constraints for our mystery number \(x\). Let \(c_1\) and \(c_2\) be the factors required to multiply \(x\) to get our dividend less the remainder.
\[c_1x=62-2\Rightarrow \underline{c_1x=60}\]\[c_2x=80-5 \Rightarrow \underline{c_2x=75}\]Let us expand the prime factorization of the right-hand sides of both of the underlined equations. The more common factors we can use to reserve for \(x\), the bigger it will be. This is the same as finding the least common factor between \(60\) and \(75\). The common prime factors have been underlined.\[c_1x=2\cdot 2\cdot \underline 3 \cdot \underline 5\]\[c_2x=\underline 3\cdot \underline 5 \cdot 5\]Thus, the biggest (presumably whole) number that \(x\) can be is \(x=3\cdot 5 = \boxed{15}\).