# Question #d5a5c

May 5, 2016

$x = 3$

#### Explanation:

Note that as $4 \equiv - 1 \text{ (mod 5)}$ we have

${2}^{2010} \equiv {\left({2}^{2}\right)}^{1005} \equiv {4}^{1005} \equiv {\left(- 1\right)}^{1005} \equiv - 1 \text{ (mod 5)}$

Thus we are actually searching for the least $x \in {\mathbb{Z}}^{+}$ such that

$3 x \equiv - 1 \text{ (mod 5)}$

From here, we can simply iterate $x$ starting from $x = 1$.

$3 \left(1\right) \equiv 3 \cancel{\equiv} - 1 \text{ (mod 5)}$

$3 \left(2\right) \equiv 6 \equiv 1 \cancel{\equiv} - 1 \text{ (mod 5)}$

$3 \left(3\right) \equiv 9 \equiv - 1 \text{ (mod 5)}$

Therefore the least such positive integer $x$ is $x = 3$