Question

There are two 3-digit numbers $n$ having the property that $n$ is divisible by 11 and $\dfrac{n}{11}$ is equal to the sum of the squares of the digits of $n$. Find both values of $n.$ You may submit them in either order.

Answer 1

550 and 803

Explanation:

Let the number be N = abc

Then:

• abc/11 = (a² + b² + c²) ⇒ abc = 11(a² + b² + c²)

There are:

• 1000 - 100 = 900 3-digit numbers

Out of 900 numbers:

• 1000/11 - 100/11 = 81 are divisible by 11

The easiest and simplest way is to try them all.

The trial reveals 550 and 803, so this is the answer.