Medium

The product of two positive integers $x$ and $y$ is 6760 and their HCF is 13. How many such pairs $(x, y)$ can be formed?

✅ Correct Option: 1

The HCF is 13, so write $x=13a$ and $y=13b$, where $a$ and $b$ share no common factor. $xy=6760$ $\Rightarrow 169ab=6760$ $\Rightarrow ab=\dfrac{6760}{169}=40$ The coprime factor pairs of 40 are $(1,40)$ and $(5,8)$. Since $(x,y)$ is an ordered pair, their reverses must also be counted. | $(a,b)$ | $(x,y)$ | |---|---| | $(1,40)$ | $(13,520)$ | | $(40,1)$ | $(520,13)$ | | $(5,8)$ | $(65,104)$ | | $(8,5)$ | $(104,65)$ | Therefore, 4 such ordered pairs $(x,y)$ can be formed.

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