Please help me with this discrete math question Thank you Qu

Please help me with this discrete math question. Thank you!

Question 6-

(i) Prove that the sum of two consecutive integers is an odd integer.

(ii) The square of an even number is an even number.

Solution

Solution i)

Let n be an odd number . Then both (n+1) and (n-1) will be even,
So we know that (n+1)/2 and (n-1)/2 are integers.
Now (n+1)/2 and (n-1)/2 are consecutive as (n-1)/2 + 1 = (n-1)/2 + 2/2 = (n+1)/2

Then, (n-1)/2 + (n+1)/2 = 2n/2 = n which is an odd number

So, (n-1)/2 and (n+1)/2 are consecutive integers which sum to n.

Solution ii)

Let n=2m is an even integer where m is an integer. Since n=2m , thus

n²= (2m)²= 4m² = 2 (2m)²

Here, (2m)² is an integer as m is an integer so let p =(2m)^2. Then n² = 2p which is even

Hence, the square of an even number is an even number.