Odd and Even Numbers

Ever stayed at a hotel? Room 1, 3, 5... run down one side of the corridor and 2, 4, 6... down the other. Every whole number belongs to one of these two camps: odd or even. The split takes one glance at the last digit, yet it keeps ruling the worlds of adding, subtracting and multiplying.

Spot odd or even at a glance

A number ending in 0, 2, 4, 6 or 8 is even; one ending in 1, 3, 5, 7 or 9 is odd. Big numbers are no scarier: 3478 ends in 8, so it is even; 20719 ends in 9, so it is odd.

Why is the last digit enough? The tens, hundreds and every higher place count things that pair up perfectly. Being even means "everything pairs off": 12 socks make 6 complete pairs, while 13 socks leave one lonely leftover after 6 pairs. Zero is even too — $0 \div 2 = 0$ , nothing left over.

Odd and even in adding and subtracting

even + even = even: $4 + 6 = 10$ ;
odd + odd = even: $7 + 9 = 16$ — two leftovers finally make a pair;
even + odd = odd: $8 + 5 = 13$ — pairing cannot finish the job, one stays single.

Subtraction follows the same script: $16 - 9 = 7$ , even minus odd gives odd. So you can decide the parity of $1024 + 317$ without computing: even plus odd must be odd (and indeed the sum is 1341).

Checking without computing

After working out $3907 + 42 = 3949$ , how do you self-check in seconds? Look at parity: odd plus even must give an odd result, and 3949 ends in 9 — it does. If your answer ended in an even digit, something went wrong. Parity is the cheapest lock on the ledger.

Odd and even in multiplying

Multiplication has a one-line rule: if any factor is even, the product is even; only all-odd factors give an odd product.

$23 \times 24$ : since 24 is even, the product must be even (it is 552);
$3 \times 5 \times 7$ : all odd, so the product must be odd (105).

The reason lives in factors and multiples: every even number carries a factor of 2 and hands it to the product. Notice this is the opposite of adding — odd + odd is even, yet odd × odd is odd. The pair is easy to mix up.

Parity in a triple product

Is $7 \times 15 \times 22$ odd or even? No need to multiply: 22 is even, so the product must be even. It really is 2310, ending in 0 — checks out. Likewise odd × odd × odd is always odd: $3 \times 5 \times 7 = 105$ .

The computer's trick

In binary the test is even faster: last bit 1 means odd, last bit 0 means even. However large the number, a computer only ever looks at that one digit.

Odd and even at work

Pairing up in gym class: 26 students form exactly 13 pairs; 27 students fill 13 pairs and one person must sit out — because 27 is odd. Wherever "two by two" matters — handing out gloves, sharing tables, forming doubles teams — counting parity first tells you whether anyone will be left over.

Room numbers work the same way: odds on one side, evens on the other, and neighbors on the same side differ by 2. Looking for room 1088? It ends in 8, an even digit — walk down the even side and you cannot miss it.

Check yourself

Quick quiz

1. Without computing, is 3907 + 42 odd or even?
2. Which result is guaranteed to be odd?
3. 27 students pair up two by two for a game. What happens?