UPSC Prelims 2025 CSAT, question 5

Solution.

Answer: (c) 8

1. Dividing $N = p + q + r$ by $N$ gives $\frac{p}{N} + \frac{q}{N} + \frac{r}{N} = 1$, or $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1$, where $a = \frac{N}{p}, b = \frac{N}{q}, c = \frac{N}{r}$ are distinct positive divisors of $N$ greater than $1$.
2. The only set of three distinct integers greater than $1$ whose reciprocals sum to $1$ is $\{2, 3, 6\}$ (since $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1$), so $N$ has this property if and only if $N$ is a multiple of $6$ (with $p = \frac{N}{2}, q = \frac{N}{3}, r = \frac{N}{6}$).
3. The multiples of $6$ below $50$ are $6, 12, 18, 24, 30, 36, 42,$ and $48$, which gives $8$ numbers.