Picking Parity

Class 7 Maths · Chapter 6: Number Play · NCERT Ganita Prakash section 6.2

Learn

Kabir has number cards 1, 3, 5, 7, 9, 11, 13 and wants 5 cards that add to 30. Instead of trying every choice, think in pairs. An even number can be arranged in pairs with nothing left over; an odd number always has one left over.

Even + even is even. Odd + odd is even: the two leftovers pair up! But 3 odd numbers leave one over: odd. So 5 odd numbers always add up to an odd number, and 30 is even: Kabir's puzzle is impossible. Being even or odd is called parity.

odd + odd = even · 5 odds → odd

Meher is exactly one year older than Neel, so their ages are consecutive, one even and one odd. Their sum is odd, so it can never be 112. And for a grid of r rows and c columns, r × c is odd only when both r and c are odd: 27 × 13 is odd, 42 × 78 is even.

even + odd = odd · odd × odd = odd

Parity of expressions: 3n + 4 can be odd (n = 3 → 13) or even (n = 8 → 28). But 100p and 48w − 2 are always even, and 2m + 1 is always odd. The nth even number is 2n, and the nth odd number is one less: 2n − 1. So the 100th odd number is 199.

nth even = 2n · nth odd = 2n − 1

Explore: even or odd?

1. Sort each result by its parity.

Remember

• Even numbers pair up fully; odd numbers have one left over.
• An even count of odd numbers adds to even; an odd count adds to odd. The nth odd number is 2n − 1.

Quick check

A piggy bank has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins. Could the total be ₹205?

1. No: the total must be even
2. Yes
3. Only if there are no ₹10 coins
4. Only with 1 coin of each

Think about it

Is the sum 1 + 2 + 3 + … + 100 even or odd?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.