SOLUTION: a=b+2 a+b=axb show that a and b are not integars

Question 131519: a=b+2
a+b=axb
show that a and b are not integars

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
I'm assuming that your 'axb' really means 'a times b'

So:
and

Substitute into the second equation:

Add -2b to both sides:

So: or , therefore or

1. Assume that is a rational number, meaning that there exists an integer and an integer such that .
2. Then can be written as an irreducible fraction such that and are coprime integers and .
3. It follows that and . ()
4. Therefore is even because it is equal to . is necessarily even because it's divisible by 2�that is, � and numbers divisible by two are even by definition.)
5. It follows that must be even as (squares of odd integers are also odd, referring to b) or (only even numbers have even squares, referring to a).
6. Because is even, there exists an integer k that fulfills: .
7. Substituting from (6) for in the second equation of (3): is equivalent to is equivalent to .
8. Because is divisible by two and therefore even, and because , it follows that is also even which means that b is even.
9. By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).
Since there is a contradiction, the assumption (1) that is a rational number must be false. The opposite is proven: is irrational. (courtesy Wikipedia)

Since all integers can be represented as the quotient of integers, all integers are rational. Since has been proven to be irrational, cannot be an integer. Since all integers can be represented as the sum of 2 and some other integer and has been shown to be other than an integer, and are also not integers.

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