Prove that if n is a positive integer. then n is even if and only if 7n + 4 is even.

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

  1. 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


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