[maskedmelonpai]

well, okay, butt i ain't actually written any proofs yet, so this will be my first time (o^^o)

Proposition: The Lease Common Multiple of a coupla numbers x & y, is they product divided by they Highess Common Factor, that is:

∀(x,y)Ǝlcm.lcm = (x *y)/hcf(x,y)

1. both numbers x & y are liek the product of a buncha smaller ones that're called
"primes." that to say: [i]for every number (n), there exist a set (s) of smaller numbers, whomsts product (p) = (x). another words:

∀nƎs.P(s) = n

i think this pretty obvious and not really sure how or why to prove it ATM, so just gonna say it a lemma an move on.

2. The LCM of a coupla numbers is the littlest number that both the numbers can fit into evenly if you made a buncha copies of each. to find the LCM of a pair, we take an get ridda all the overlapping factors between them, such that there are enough prime factors left over that we can still make each number, but that's it. THEN we multiply all those factors together.

3. The hcf of two numbers is the sum of all the overlapping prime factors between the two numbers. That is to say that for every set (p) of prime factors of a product, there exist a set (o) of overlapping prime factors between the two multiples that the product come from. another words:

∀pƎo. HCF(x,y) = f(o) where f is product of o


An then we try it out:



Let x = 6, y = 4;
F(x) = 2,3
F(y) = 2,3
LCM(x,y) = 2*2*3 = 12 *cause we need two 2s to make a four and one 2 and a 3 to make a six.
HCF(x,y) = 2

LCM(x,y) = (x * y)/HCF(x,y)
=(6*4)/2
=24/2
=12

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