Question

The sum of the squares of two consecutive natural numbers is `365`. What are the two numbers?

Answer 1

`n = 13`

Explanation:

Making the equation

`365 = n^2+(n+1)^2` and solving for `n` we obtain

`n = -14` and `n = 13` so we follow with `n = 13`

Answer 2

There are two solutions: `(13 and 14)` and `(-14 and -13)`, however negative numbers are not 'natural numbers', so the solution is `13` and `14`.

Explanation:

Call the first number `n`, then the next consecutive number will be `n+1`. We will have:

`n^2+(n+1)^2=365`

Expanding:

`n^2+(n^2+2n+1)=365`

Collecting like terms:

`2n^2+2n+1=365`

Subtract 365 from each side to get a quadratic equation in standard `ax^2+bx+c=0` form:

`2n^2+2n-364=0`

There are a number of ways of solving quadratic equations, but my favourite is the quadratic formula:

`n=(-b+-sqrt(b^2-4ac))/(2a)=(-2+-sqrt(2^2-4xx2xx(-364)))/(2xx2)`

`=(-2+-sqrt(4+2912))/(4)=(-2+-54)/(4)`

The possibilities for `n` are `-14` or `13`.

This is intriguing: let's not dismiss the negative numbers too quickly, because remember, we are squaring them. Let's test 13 first:

`13^2+14^2=365`

Now let's check `-14`:

`-14^2+ -13^2=365`

There really are two sets of answers, but as noted above, the term 'natural numbers' does not include negative numbers, so only one of them answers the question asked.