1. For any integer n, determine the parity of

n² + 3n + 2

2. If x is an odd integer, what can be said about

x³ + x

3. Let n be any integer. Determine whether

n(n + 1)(n + 2)

is odd or even. Justify your answer.

4. If a is even and b is odd, determine the parity of

a² + ab + b²

5. Determine whether

2n² + 5n + 7

is always even, always odd, or sometimes even and sometimes odd.

6. The product of four consecutive integers is always divisible by:

• (a) 2
• (b) 4
• (c) 8
• (d) 16

7. The sum of five consecutive integers is even.

What can you conclude about the middle integer?

8. Can the sum of three odd integers be even?

Explain your reasoning.

9. The product of seven consecutive integers is taken.

Is the product always divisible by 8? Give reasons.

10. Five consecutive integers are added together.

Under what condition is their sum odd?

11. A number is formed by multiplying three even numbers and four odd numbers.

What is the greatest power of 2 that always divides the product?

12. If x and y are integers and

x + y is odd,

what can you say about the parity of x and y?

13. Is it possible for the product of two integers to be odd if their sum is even?

Explain.

14. The sum of nine consecutive integers is odd.

What can you conclude about the middle integer?

15. Prove or disprove:

n² + n is always even for any integer n.