n where it gives certain remainder for certain number

Question

I am studying for GRE and need help with following question

When the positive integer n is divided by 3, the remainder is 2 and when n is divided by 5, the remainder is 1. What is the least possible value of n?

Answer says 11, but gives no explanation.

I tried to solve this, i.e.

$n = 3x + 2$

$n = 5y + 1$

$3x + 2 = 5y + 1$

$3x - 5y + 1 = 0$

$1 = 5y - 3x$

I'm not sure how to proceed. Please guide.

Answer 1

You have the two congruences $n\equiv_3 2$ and $n\equiv_5 1$. We know that the answer should look like $5k+1$ for some integer $k$, and we see that if $k=0$, we get $1$, when $k=1$ we get $6$, and when $k=2$ we get $11$. In trying $1$ and $6$ we see that it doesn't work. But it does for $11$.