We want to prove that for any positive integer k rational nu

We want to prove that for any positive integer k, rational number). k is either integer or irrational (can’t be non-integer )

a. We saw before that 2|n2 implies 2|n for any n. Let’s call a number k round if k|n2 always implies k|n. Describe the set R of all round numbers in terms of prime factorizations of those numbers.

b. Show that if k is round and k = 1, then k is irrational.

c. Show that if m is a positive integer, m can be represented as a product m = nk, where n is a positive integer and k is a round number.

d. Show that if m is a positive integer, m is either integer or irrational.

Solution

b)

, let p1k be rational p1k=ab with (a,b)=1or p= akbk,

since p is a prime hence integer we have b=1and p= ak, and if k>1then p will be divisible by more then two numbers, hence it will not be prime, a contradiction, therefore p1k is irrational for k>1and p be a prime.

Here from above example , if we write k in lace of p

And k=1/2 here

So k^.5 is irrational when k not equal to 1.

d)

Let n be a positive integer such that there is no m such that n=m2. Suppose nn is rational. Then there exists p and q with no common factor (beside 1) such that

n=p/q

Then

n=p2/q2.

However, n is an positive integer and p and q have no common factors beside 1. So q=1This gives that

n=p^2

Contradiction since it was assumed that nm^2 for any m.

If we want to prove n in place of m

Just replace m with n in the above process.