Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 104 / 5 / a

In a direct product of groups, $(a,b{)}^r=(a^r,b^r)$. Thus a finite order for $(a,b)$ makes both coordinate orders finite. Conversely, if their orders are $p$ and $q$, the least positive exponent annihilating both is their least common multiple. Hencewhen both orders are finite; if either coordinate has infinite order, so does $(a,b)$.