1.prove that if n is a positive integer, then n is evenif and
only if 7n + 4 is even

2. prove that if x is rational x doesnt equal 0 , then 1/x
isrational.

thanks

## Answer

- Prove if n is even, 7n+4 is even:

if n is even, then we can say n=2k for some positive integerk.

so 7n+4=14k+4=2(7k+2), so we see that 7n+4 is a multiple of2, so it’s even

now let’s prove the inverse: prove if n is odd, then7n+4 is odd

If n is odd, then n=2k-1 for some positive integer k

7n+4=7(2k-1)+4=14k-7+4=14k-3=2(7k-1)-1

So, because 7n+4 is one less than a multiple of 2, it mustbe odd

i don’t really know how to prove 1 but

for 2. if x is rational then it can be expressed as x=a/b

then 1/x= b/a