Number Play · Lesson 2 of 10
Picking Parity
“Use pairs to explain which sums and differences are possible.”
• Explain even and odd numbers using pairs and leftovers. • Predict the parity of sums with several odd and even addends. • Use parity to rule out impossible totals without testing every choice. • Determine the parity of differences between even and odd numbers.
Can five odd cards total 30?
Five boxes must each contain one number card, and the total must be 30. If every available card has an odd number, do we need to inspect all choices? A pattern shared by all odd numbers answers the question before any trial-and-error search.
Parity is the property of a whole number being even or odd. An even number can be separated into pairs with nothing left over; an odd number leaves one unpaired item.
Any number of even collections can be combined into pairs, so their sum is even. Each odd collection contributes one leftover. Two leftovers make a pair; three leftovers leave one after pairing two of them. Consequently, an even number of odd addends has an even sum and an odd number of odd addends has an odd sum. Additional even addends do not change that result.
Problem
Can five odd cards add to 30?
- 1.Each of the five odd numbers contributes one unpaired item.
- 2.Four of the five leftovers pair up; one remains. The sum of five odd numbers is odd.
- 3.Thirty is even, so no selection of five odd cards can give 30.
Consecutive numbers and everyday totals
As we count 1, 2, 3, 4, …, even and odd alternate. Two consecutive ages therefore contain one even and one odd number. Their sum must be odd. If siblings born exactly a year apart are both celebrating a birthday, an even proposed age sum such as 112 cannot fit their two consecutive ages.
Problem
Lakpa has an odd count of ₹1 coins, an odd count of ₹5 coins, and an even count of ₹10 coins. Could the value be ₹205?
- 1.Each ₹1 coin contributes an odd amount, so an odd count of them has an odd total. An odd count of ₹5 coins also has an odd total.
- 2.Each ₹10 coin contributes an even amount, so its group has an even total.
- 3.Odd + odd + even is even. Since ₹205 is odd, the claimed total is impossible.
What about subtraction?
Subtraction changes a total, but the pair-and-leftover reasoning still works. Removing an even number preserves the original parity. Removing an odd number switches it. Check the pattern with actual numbers first, then use it for any whole-number example where the subtraction makes sense.
| Difference | Example | Parity |
|---|---|---|
| even − even | 12 − 4 = 8 | even |
| odd − odd | 13 − 5 = 8 | even |
| even − odd | 12 − 5 = 7 | odd |
| odd − even | 13 − 4 = 9 | odd |
Problem
Without calculating 857 − 246, decide whether the result is odd or even.
- 1.The last digit 7 shows that 857 is odd; the last digit 6 shows that 246 is even.
- 2.Odd − even is odd. Subtracting confirms 857 − 246 = 611, an odd number.
An even sum does not mean that every addend is even. Two odd numbers also add to an even number.
Quiz
What is the parity of the sum of five odd numbers?
What is even + odd + odd + even?
Two consecutive whole numbers always sum to a number of which parity?
What is the parity of odd − even?
Why can five odd cards not total 30?
Practice Problems
- Predict the parity of the sum of three odd numbers and four even numbers. Explain using leftovers.
- Can six odd numbers sum to 101? Give a parity argument.
- Make three different pairs of consecutive numbers and check the parity of each sum.
- Decide the parity of 9,321 − 2,104 without carrying out the full subtraction.
- A light is initially on and its switch is toggled 77 times. Is it on or off? Explain the two-toggle pattern.
- A claim says that an even sum must have only even addends. Give a counterexample.
Key Takeaways
• Even numbers form complete pairs; odd numbers have one leftover. • The parity of a sum depends on how many odd addends it has. • Consecutive whole numbers have opposite parity, so their sum is odd. • Subtracting an even number preserves parity; subtracting an odd number changes it.