Question

Question #d5a5c

Answer 1

`x=3`

Explanation:

Note that as `4-=-1" (mod 5)"` we have

`2^2010 -=(2^2)^1005 -= 4^(1005) -= (-1)^1005 -= -1" (mod 5)"`

Thus we are actually searching for the least `x inZZ^+` such that

`3x -= -1" (mod 5)"`

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

`3(1) -= 3 cancel(-=) -1" (mod 5)"`

`3(2) -= 6 -= 1 cancel(-=) -1" (mod 5)"`

`3(3) -= 9 -= -1" (mod 5)"`

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