The greatest common divisor of positive integers m and n is 8. The least common multiple of m and n is 112. What is the least possible value of m + n?

Question

anonymous · Answer

What I thought was that the l.c.m. multiplied by the g.c.d. of two numbers is equal to m*n, when I multiplied that out I got 869, did prime factorization and got 128*7.... where am I wrong?

anonymous · Answer

hello?

rizags · Answer

hello?

rational · Answer

m = 8m'
n = 8n'

mn = 8*112
m'n' = 14

so (m', n') =  (1, 14), (2, 7), (7, 7)
the least value of m' + n' is 2+7 = 9
so the least value of m+n is 8*9 = 72

anonymous · Answer

but, what is m' and n' ? what is the meaning of ( ' )?

rizags · Answer

lemme do a bit of casework on this one

rizags · Answer

alrite, so to begin, we know that $$m, n \le 112$$ because their least common multiple is 112, and multiples of positive integers are always greater than or equal to themselves

rizags · Answer

yea that guy rational up there, is right

rizags · Answer

his solution is very elegant, but heres a simpler one

rizags · Answer

to begin, list all factors of 112 in ascending order for me

rizags · Answer

heloooooo? could you do that ^^