Question

N is a positive integer.

(i) Explain why n(n - 1) must be an even number.

(ii) Explain why 2n + 1 must be an odd number.

Answer 1

() = -1

' { -1} -1

() ,

× -1= (-1 )

() = 3

2+1=3

3

Answer 2

(i) When we expand n(n-1), we get n^2 - n. Notice that if n is even, then n^2 is also even because it is the product of two even numbers. Similarly, if n is odd, then n^2 is odd because it is the product of two odd numbers. However, regardless of whether n is even or odd, the term "-n" is always odd. Therefore, n^2 - n is always even, which means that n(n-1) must be an even number.

(ii) If we add 1 to any even number, we get an odd number. Therefore, if 2n is an even number, then 2n + 1 must be an odd number. Alternatively, we can use the definition of an odd number, which is a number that cannot be divided by 2 without leaving a remainder. If we divide 2n by 2, we get n with no remainder, which means that 2n is an even number. If we add 1 to an even number, we get an odd number. Therefore, 2n + 1 must be an odd number.