KA-0002Parity of numbers (odd/even behavior)
3 points, difficulty 1 of 3Level 3–4about 75sAna adds up the first seven odd numbers: 1 + 3 + 5 + 7 + 9 + 11 + 13. Is her total odd or even, and what is it?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Pair the numbers from the outside in.
2The strategy
Odd plus odd is even, so pairs of odd numbers give an even total. Seven numbers make three pairs with one number spare, and that spare number decides the parity.
3The full solution
Pair 1 with 13, 3 with 11, 5 with 9. Each pair makes 14, giving 42, and the 7 in the middle is left over. 42 + 7 = 49, which is odd because an even total plus one odd number is odd.
Solution
The reliable way
Pair the first with the last: 1 + 13 = 14, 3 + 11 = 14, 5 + 9 = 14. That is three pairs making 42, and 7 sits alone in the middle. The total is 42 + 7 = 49. Since 42 is even and 7 is odd, the answer is odd. The transferable idea: pairing from the outside in turns a long sum into a short multiplication.
The elegant way
The sum of the first n odd numbers is always n squared, so seven odd numbers add to 49 - and a square of an odd number is odd.
Why this is on the test: It rewards pairing over grinding, and it punishes the belief that a pile of odd numbers must add to something odd.