Problem 8 · Sum equals product
The only possible consecutive numbers are 1, 2 and 3.
Integer answer, at most 4 digitsThe sum of 3 consecutive natural numbers equals their product. What is the sum of their cubes?
Copa Cangur · SCM
Easy
Closed answer
Reasoned solution
Key idea: write the consecutive numbers as $n-1$, $n$, $n+1$: the sum is $3n$ and the product $n(n^{2}-1)$.
$$n(n^{2}-1) = 3n \;\Longrightarrow\; n^{2} - 1 = 3 \;\Longrightarrow\; n = 2.$$
The numbers are $1, 2, 3$ (indeed $1+2+3 = 6 = 1 \cdot 2 \cdot 3$). The sum of cubes:
$$1^{3} + 2^{3} + 3^{3} = 1 + 8 + 27 = 36.$$
Answer: 36