LCM New Year

Number Theory Level 3

Consider all possible natural numbers $a, b, c, x, y,$ and $z$ such that
1) $LCM(a,b,c) = LCM(x,y,z) = 2015,$
2) both $a+b+c$ and $x+y+z$ are perfect squares,
3) $a, b, c$ are distinct,
4) $x, y, z$ are distinct.

Find the maximum value of $a - b - c - x - y - z.$

94 91 93 92

2 solutions

Srikanth Kummara
Dec 6, 2014

Here to have maximum value a should be maximum possible factor...that is greatest divisor....it is 155.....from here on rearranging we get two sets... 155;13;1 and 31;13;5.....hence solved

Patrick Corn
Dec 3, 2014

I don't think this is correctly stated. The only triples that work are permutations of $(155,13,1), (31,13,5),$ and $(155,65,5)$ . It's not possible to find two triples with six distinct natural numbers as the problem asks for. But if you take the first two with $a = 155$ , you get $92$ for an answer.

Thanks. I have removed the condition that they are distinct.

Calvin Lin Staff - 6 years, 6 months ago

I don't think that works; I assumed that $a,b,c$ were distinct and $x,y,z$ were distinct. Otherwise you could use e.g. $(a,b,c) = (2015,5,5)$ to do even better

Patrick Corn - 6 years, 6 months ago

Thanks. I've added that in.

Calvin Lin Staff - 6 years, 6 months ago

Th anks for the correction sir. Im beginner so ive got so many mistake here

Rimba Erlangga - 6 years, 6 months ago

LCM New Year

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