Question

Number Thoery by Geogre Andrews page 68. Explain the part where it says "Let ni = M/mi. Since no two of the mi have a common factor, we see that gcd(ni, mi) = 1" Why?

Answer 1

https://books.google.com/books?id=NV68AQAAQBAJ&lpg=PA58&pg=PA68#v=onepage&q&f=false

Answer 2

Wasn't that the supposition at the statement of the theorem above?

Answer 3

uhm.. no. The assumption was no two of the mi have a common factor.

M = m1 * m2 * ...* ms

ni = M / mi, the claim is gcd(ni,mi) = 1

Answer 4

It has to do with how M is set up

M = m1*m2*m3 ... *ms

and how the mi's are pairwise coprime

eg:

If i = 2, then

ni = M/mi
n2 = M/m2
n2 = m1*m3*m4*...*ms

n2 and m2 have no factors in common other than 1 because m2 and m1*m3*m4*...*ms have nothing in common other than 1

Answer 5

@jim_thompson5910 oh I see now. It has something to do with finding the greatest common factor of two numbers by prime factorization. (something I forgot I learned in grades school lol)

Answer 6

that is correct

sometimes you'll see `gcd(a,b)` to mean `the greatest common factor of a and b`

other times you may see `(a,b)` to mean the same thing

Answer 7

@jim_thompson5910 awesome. Thanks :)

Answer 8

you're welcome