Show that if n is a multiple of 3 then 2n can be broken into

Show that if n is a multiple of 3, then 2/n can be broken into the sum of two unit fractions, one of which is 1/n. Then, show that if n is a multiple of 5, then 2/n can be broken into a sum of two unit fractions, one of which is a third of 1/n.

Solution

1.) Since n is a multiple of 3, we may write n = 3m.

Now we can write 2/n = 2/(3m) = (1/n )+ x = 1/(3m) + x.

Solving for x gives x = 1/(3m) .

It follows that 2/n = (1/n) + (1/(3m)) , where n = 3m.

2.) We write n = 5m,

and then have 2/n = 2/(5m) = (1/(3n)) + x = (1/(15m)) + x,

which gives x = 5/(15m) = 1/(3m) .

Thus 2/n = (1/(3n)) + (1/(3m)) , where n = 5m