# Integer puzzle

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• November 14th 2009, 04:12 AM
Bacterius
Integer puzzle
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 ...
• November 14th 2009, 11:02 AM
Defunkt
Are you sure about this?

since $a, $a^2 and so $a^2 \equiv a^2 (mod \ b^2)$ and also $a^2 \equiv a (mod \ b)$ but $a 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...
• November 14th 2009, 01:47 PM
Bacterius
Quote:

Originally Posted by Defunkt
Are you sure about this?

since $a, $a^2 and so $a^2 \equiv a^2 (mod \ b^2)$ and also $a^2 \equiv a (mod \ b)$ but $a 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 !