BdMO National Secondary 2010#6
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- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
Find ,with proof,all the perfect squares each of which is the product of four consequetive odd natural numbers.
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- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
Re: BdMO National Secondary 2010#6
There are no such perfect square.
Proof:
Let the numbers are $(k-3),(k-1),(k+1),(k+3)$
So,there product$ =k^4-10k^2+9$
This product will be a perfect square if and only if $k=0$ but then two of the numbers will be negative.
So,there are no such perfect square which is product of four consequetive odd natural numbers.
[Q.E.D]
Proof:
Let the numbers are $(k-3),(k-1),(k+1),(k+3)$
So,there product$ =k^4-10k^2+9$
This product will be a perfect square if and only if $k=0$ but then two of the numbers will be negative.
So,there are no such perfect square which is product of four consequetive odd natural numbers.
[Q.E.D]