# iff proof

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• March 11th 2008, 11:03 PM
asw-88
iff proof

Prove that n is a square iff the exponent of every prime number occuring in the factorisation of n is even.

Hence prove that the square root of any natural number is either an integer or an irrational number.

Any help would be greatly appreicated I don't know anyone else who can.
• March 15th 2008, 07:34 AM
Moo
Hello,

Write the decomposition into prime factors of n.

$n = \prod_{i=1}^k p_i ^{\alpha_i}$

So $n^2 = \prod_{i=1}^k p_i ^{\alpha_i} \prod_{i=1}^k p_i^{\alpha_i} = \prod_{i=1}^k [p_i^{\alpha_i} \times p_i^{\alpha_i}]$

As $a^b a^c = a^{b+c}$, the previous expression equals to :

$\prod_{i=1}^k p_i^{\alpha_i + \alpha_i} = \prod_{i=1}^k p_i^{2 \alpha_i}$

Hence the exponent of every prime number occuring in the factorisation of n is even because multiple of 2.

And as it's equalities, there is equivalence.
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