Mathematics
I spend my free time in learning Mathematics and solving interesting problems. While solving problems, I come across many interesting facts and theorems which I try to prove. I have collected most of the documents in the mathematics directory.
This page contains a brief explanation of the documents with a link to them.
Factors
Some results on factors of positive integers.
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Unique mods in an LCM interval
The L.C.M. of 2 and 3 is 6. Let us take an integer in the
closed interval [0, 5], say, 5. 5 leaves a remainder of 1
and 2 on dividing by 2 and 3 respectively. There is no other
integer in this interval which will leave such remainders.
In general, for any two positive integers m and n there can
not be two integers in the closed interval [a, a + lcm(m, n)
- 1] such that they leave the same remainders on dividing by
m and n.
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Integer triangles with fixed longest side
The possible number of triangles with integer sides with a
given length of the longest side.
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Integer triangles with fixed perimeter
The possible number of triangles with integer sides with a
given sum of the three sides.
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