Using algebra to demonstrate an argument

Algebra can be used to show the properties of expressions and demonstrate when different expressions are equivalent. Rules can be used that apply to sets of numbers, such as odd numbers and even numbers, rather than applying just to individual numbers.

Rules of odd and even numbers

The following rules apply for any even or odd numbers:

even + even = eveneven x even = even
odd + odd = evenodd x odd = odd
even + odd = oddeven x odd = even
odd + even = oddodd x even = even

Examples can be used to demonstrate that these rules are true, although mathematical proof would be required to show that the rules are true in all cases. Finding one example where a rule does not work (called a counter-example) is enough to show that the rule does not always work.

Example

If p is an even number, show that (p + 1) \times (p + 5) is odd.

If p is an even number, then p + 1 and p + 5 will both be odd because even + odd = odd.

(p + 1) \times (p + 5) is therefore odd because odd x odd = odd.

Example

Jack says, “Every integer that ends in 3 is a prime number”. Find an example to show that Jack’s statement is not correct.

3, 13 and 23 are all prime numbers. However, 33 is not prime because 3 × 11 = 33, so Jack’s statement is not correct.

Question

If t is odd, explain why 5t + 3 is even.

If t is odd then 5t is also odd, because odd × odd = odd.

5t + 3 is therefore even because 5t + 3 is odd + odd = even.