Question

N is a two-digit positive even integer where the sum of the digits is 3. If none of the digits are 0, what is N?

Answer 1

`12`

Explanation:

If `N` is a two-digit positive number, where the sum of the digits is `3`, the only two possibilities for `N` is:

`12` and `30`

But since none of the digits are `0`, that excludes `30` from being an option, and so the answer is `12`.

Answer 2

12

You can get this quite easily through just thinking about it, but I'll demonstrate an algebraic approach.

Explanation:

If `N` is a two digit number, we can write this as `N=10x+y`, where `x` and `y` are positive non-zero integers less than 10.
Think about it - every 2 digit number is 10 times something (your 10s digit) plus another number.

We also know that `N` is even i.e. it is a multiple of 2. This means that `y` must be equal to `2xx"something"`. If we let this something be another variable `u`, `y=2u`

`:. N=10x+2u`
where `x in NN, 0< x <10` and `u in NN, 0< u<5`

We know that we are looking for `x+y`, or `x+2u`

`x+2u=3`

We can use a graph to find all the solutions that satisfy our previous limits on x and u.

graph{x+2y=3 [-0.526, 3.319, -0.099, 1.824]}

The only integer solutions in this range are `x=1` and `u=1`

`:. N=10(1)+2(1)`
`N=10+2`
`N=12`