Two are relatively prime iff. their LCM=ab.

1. Assume a and b are relatively prime.

2. Assume .

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- Jun 20th 2010, 03:59 PMdwsmithTwo positive Z are relatively prime iff. their LCM=ab
Two are relatively prime iff. their LCM=ab.

1. Assume a and b are relatively prime.

2. Assume .

- Jun 20th 2010, 04:05 PMundefined
- Jun 20th 2010, 04:07 PMdwsmith
- Jun 20th 2010, 04:19 PMundefined
- Jun 20th 2010, 04:21 PMdwsmith
- Jun 20th 2010, 04:46 PMundefined
Ohh okay it makes sense now.

My first thought is to use prime factorisations. I think the most convenient notation for prime factorisation is to use an infinite product and allow exponents to equal 0. This has the added advantage that we don't have to treat 1 as a special case.

where for simplicity we let

We have that

We also have that

So we have which is true if and only if either or , or both. Can you show how this last part is equivalent to ?

(Fixed a small typo) - Jun 20th 2010, 04:51 PMdwsmith
- Jun 20th 2010, 04:57 PMundefined
Well consider the integer 126. The typical way to express the prime factorisation is

But if we instead write

this gives us flexibility when comparing the prime factorisations of two different integers, because we can just match up each corresponding index without having to use notation to indicate the highest non-zero index, etc. - Jun 20th 2010, 05:04 PMdwsmith
- Jun 20th 2010, 05:05 PMundefined
Well if we're allowed to use that then the direction also follows easily.

Assume

Then

Edit: Ah, you just changed your first post and added another post saying you changed it. Looks good.