Integer puzzle

**Bacterius** · November 14th 2009, 03:12 AM

Here's a little puzzle for you guys.

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

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

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

**Defunkt** · November 14th 2009, 10:02 AM

Are you sure about this?

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

**Bacterius** · November 14th 2009, 12:47 PM

Originally Posted by Defunkt

Are you sure about this?

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

Yes I am sure, but did I say there was an answer ?

You solved the puzzle though ... I will do harder next time !

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