Question

find the number of ordered polis (m,n) of positive integers m and n such that m+n=190 and m and n are relatively prime.?
​

Answer 1

Answer:

If 1/m + 1/ n = 1/4 then

4(m+n) = m*n (A)

And we will show that one of m and n is divisible by 4.

Now, since 4 | m*n then either 4|m, 4|n or m and n are both even, but if m and n are both even, then m+n is even, which , from (A) gives 8|m*n, which means one of them is divisible by 4.

We can assume, then, that 4|m (say). Put m = 4*x, so

4(4x+n) = 4*x*n, I.e. 4x + n = x*n, I.e. 4*x = (x-1)n

Now x and x-1 are coprime, so if x-1 | 4*x then x-1|4

So, x-1 = 1, 2 or 4

X=2, 3, 5

M= 8, 12, 20

And we find all of these work:

1/8 + 1/8 = 1/12 + 1/6 = 1/20 + 1/5 = 1/4

So there are 5 ordered pairs: 5,20; 6,12; 8,8; 12,6; 20,5