# 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?

Jun 3, 2018

$12$

#### 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 $12$.

Jun 3, 2018

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 $N = 10 x + y$, 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 $2 \times \text{something}$. If we let this something be another variable $u$, $y = 2 u$

$\therefore N = 10 x + 2 u$
where $x \in \mathbb{N} , 0 < x < 10$ and $u \in \mathbb{N} , 0 < u < 5$

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

$x + 2 u = 3$

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 $u = 1$

$\therefore N = 10 \left(1\right) + 2 \left(1\right)$
$N = 10 + 2$
$N = 12$