Number Play · Lesson 3 of 10
Parity in Grids and Expressions
“See how parity can be predicted for products and algebraic expressions.”
• Predict product parity from the parity of its factors. • Determine when an expression is always even, always odd, or variable. • Use 2n and 2n − 1 to identify even and odd numbers by position. • Distinguish a rule that produces some numbers of a kind from a rule that produces all of them.
Count squares without multiplying
A grid with 27 rows and 13 columns contains 27 × 13 small squares. You could multiply to count them, but parity is easier: when both dimensions are odd, an unpaired square remains after pairing. If either dimension is even, the squares can be paired across that direction.
| First factor | Second factor | Product |
|---|---|---|
| even | even | even |
| even | odd | even |
| odd | even | even |
| odd | odd | odd |
Problem
Find the parity of 27 × 13, 42 × 78, and 135 × 654 without evaluating the products.
- 1.Both 27 and 13 are odd, so 27 × 13 is odd.
- 2.Forty-two is even, so 42 × 78 is even regardless of the parity of 78.
- 3.Six hundred fifty-four is even, so 135 × 654 is even.
Expressions and parity
A letter such as n can stand for different whole numbers. To decide the parity of an expression, look at the parity of each term. For instance, 3n has the same parity as n because 3 is odd, while adding the even number 4 leaves its parity unchanged. Thus 3n + 4 is odd for odd n and even for even n.
Problem
Compare 100p, 48w − 2, and 6k + 5 when the letters are whole numbers.
- 1.100p is even because it has the even factor 100.
- 2.48w and 2 are both even, so 48w − 2 is even wherever the expression is being used.
- 3.6k is even and 5 is odd; therefore 6k + 5 is always odd.
Checking a single value of n cannot prove that 3n + 4 is always even. Compare an even n with an odd n, then reason from the factors.
Give a position to each even or odd number
The positive even sequence is 2, 4, 6, 8, … . Each term is twice its position: the 1st is 2 × 1, the 2nd is 2 × 2, and the nth is 2n. At the same position, an odd number is one less: 1, 3, 5, 7, … . This gives 2n − 1 for the nth positive odd number, where n is a positive integer.
Problem
What are the 100th positive even number and the 100th positive odd number?
- 1.Put n = 100 in the even rule: 2n = 2 × 100 = 200.
- 2.Put n = 100 in the odd rule: 2n − 1 = 200 − 1 = 199.
The rule 6k + 2 always gives an even value for a whole-number k, but it skips many even numbers: with k = 1, 2, 3 it gives 8, 14, 20. Saying “every value of this expression is even” is different from saying “it represents every even number.” The rule 2n lists every positive even number when n runs through all positive integers.
Quiz
What is the parity of odd × odd?
Which expression is always odd for whole-number n?
What is the 40th positive odd number?
Which rule lists every positive even number when n = 1, 2, 3, …?
What happens to the parity of 3n + 4 when n changes from odd to even?
Practice Problems
- Predict the parity of 51 × 77 and 51 × 78 without multiplying.
- Give one expression that is always even and one that is always odd for whole-number n. Explain each.
- Explain why 5n + 2 can be odd or even as n changes.
- Find the 37th positive even number and the 37th positive odd number.
- List the first four values of 6k + 2 for k = 1, 2, 3, 4. Name three positive even numbers it skips.
- For positive integer p, does 2p + 1 describe every positive odd number? Check the smallest one.
Key Takeaways
• A product is odd only when all its factors are odd. • An expression may have fixed parity or may change parity with its letter values. • For positive integer n, the nth positive even number is 2n and the nth positive odd number is 2n − 1. • A rule can always produce even numbers without producing every even number.