Minimum Perimeter

A triangle has all its sides as integers marked l,m,n with l>m>n.

Also given that \left\{\frac{3^l}{10^4} \right\}=\left\{\frac{3^m}{10^4} \right\}=\left\{\frac{3^n}{10^4} \right\}

Find the minimum perimeter of the triangle.

{} denotes the fractional part.

3 Answers

Shubhodip ·

What meets the eye at first look is 3^{l}\equiv 3^{m}\left ( mod 10^{4} \right ) and 3^{m}\equiv 3^{n}\left ( mod 10^{4} \right )

After this it should be solvable but i think i need more time.

Shubhodip ·

I have seen the solution. knew something i did not:)

The answer is 3003

Here goes a hint:

Prove that if ax≡1(mod m) with gcd(a,m) = 1 and d is the minimum number such that ad≡1(mod m) then d|x. Well, only the statement is required here. The proof of this is trivial by another result (which i knew).Still medium amount of work is left.

Devil ·

I've done this, so no need of the hint. It was actually from some qsns on Number Theory that I had asked kaymant sir to mail me when I was in class XII.

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