1. ## Integer puzzle

Here's a little puzzle for you guys.

Find the smallest integers $\displaystyle a$ and $\displaystyle b$ that verify :

$\displaystyle a \in N$
$\displaystyle b \in N$
$\displaystyle 1 < a < b$
$\displaystyle a^2 \equiv a$ (mod $\displaystyle b^2$)

The answer is obvious, yet you have to think of it ...

since $\displaystyle a<b$, $\displaystyle a^2<b^2$ and so $\displaystyle a^2 \equiv a^2 (mod \ b^2)$ and also $\displaystyle a^2 \equiv a (mod \ b)$ but $\displaystyle a<b^2$ so $\displaystyle a \equiv a (mod \ b^2) \Rightarrow a = a^2 \Rightarrow a = 0,1$ both of which are not possible according to your restrictions...
since $\displaystyle a<b$, $\displaystyle a^2<b^2$ and so $\displaystyle a^2 \equiv a^2 (mod \ b^2)$ and also $\displaystyle a^2 \equiv a (mod \ b)$ but $\displaystyle a<b^2$ so $\displaystyle a \equiv a (mod \ b^2) \Rightarrow a = a^2 \Rightarrow a = 0,1$ both of which are not possible according to your restrictions...