Question

8. Prove that if n is a positive integer, then n is odd if and only if 5n+ 6 is odd.

Answer 1

To prove that n is odd if and only if 5n+ 6 is odd, we need to prove two statements. If n is odd, then 5n+ 6 is odd. If 5n+ 6 is odd, then n is odd.

Step-by-step explanation:

To prove that if n is a positive integer, then n is odd if and only if 5n+ 6 is odd, we need to prove two statements:

1. If n is odd, then 5n+ 6 is odd:

If n is odd, then it can be written as 2k+1, where k is an integer. Substituting this into the expression 5n+ 6, we get 5(2k+1)+ 6 = 10k+5+6 = 10k+11. Since 10k is even, adding 11 to an even number will always result in an odd number. Therefore, 5n+ 6 is odd.

2. If 5n+ 6 is odd, then n is odd:

If 5n+ 6 is odd, it means that it can be written as 2m+1, where m is an integer. Setting 5n+ 6 equal to 2m+1, we get 5n+ 6 = 2m+1. Subtracting 6 from both sides, we have 5n = 2m+1-6 = 2m-5. Since 2m-5 is odd, 5n must also be odd. Dividing both sides by 5, we get n = (2m-5)/5. Since (2m-5) is odd, dividing it by 5 will result in a fraction, which is not a positive integer. Therefore, n must be odd.

Answer 2

In both cases, we have shown that if
`$n$` is odd, then
`$5n+6$` is odd, and if
`$5n+6$` is odd, then
`$n$` is odd. This proves that if
`$n$` is a positive integer, then
`$n$` is odd if and only if
`$5n+6$` is odd.

To prove that if
`$n$` is a positive integer, then
`$n$` is odd if and only if
`$5n+6$` is odd, we need to show both implications separately.

First, let's assume that
`$n$` is odd. This means that
`$n$` can be expressed as
`$2k+1$`, where
`$k$` is an integer.

Now, we can substitute this expression for
`$n$` in the equation
`$5n+6$`.

`$5n+6 = 5(2k+1)+6 = 10k+5+6 = 10k+11$`

We can rewrite
`$10k+11$` as
`$2(5k+5)+1$`, which is an odd number since
`$5k+5$` is an integer.

Therefore, if is odd, then
`$5n+6$` is also odd.

Next, let's assume that
`$5n+6$` is odd. This means that
`$5n+6$` can be expressed as
`$2m+1$`, where
`$m$` is an integer.

Now, we can solve this equation for
`$n$`.

`$5n+6 = 2m+1$`

Subtracting 6 from both sides, we have:

`$5n = 2m+1-6 = 2m-5$`

Now, we can rewrite
`$2m-5$` as
`$2(m-3)+1$`, which is an odd number since
`$m-3$` is an integer.

Therefore, if
`$5n+6$` is odd, then
`$n$` is also odd.