A natural number $N$ is such that it can be expressed as $N = p + q + r$, where $p$, $q$ and $r$ are…
A natural number $N$ is such that it can be expressed as $N = p + q + r$, where $p$, $q$ and $r$ are distinct factors of $N$. How many numbers below 50 have this property?
6
7
8
9
Solution
We need $N$ below 50 expressible as the sum of three distinct factors of $N$. The classic family is $N = \dfrac{N}{2} + \dfrac{N}{3} + \dfrac{N}{6}$, valid when $N$ is divisible by 6. Numbers below 50 divisible by 6: 6, 12, 18, 24, 30, 36, 42, 48 - that is 8 numbers (e.g. $6 = 3+2+1$, $12 = 6+4+2$, $18 = 9+6+3$, etc.). Hence 8 numbers have the property.