Numbers equal to the sum of the cubes of their digits

Statement

Find all three-digit numbers equal to the sum of the cubes of their digits.

Solution

We look for numbers $\overline{abc} = 100a+10b+c$ with $\overline{abc} = a^3+b^3+c^3$. There are four: $\boxed{153,\;370,\;371,\;407}.$ Check: $\begin{aligned} 153 &= 1^3+5^3+3^3 = 1+125+27, \ 370 &= 3^3+7^3+0^3 = 27+343+0, \ 371 &= 3^3+7^3+1^3 = 27+343+1, \ 407 &= 4^3+0^3+7^3 = 64+0+343. \end{aligned}$