Numbers & the Number System • Topic 8 of 9

Odd and Even Numbers

Parity is whether a number is even or odd, and it is decided by one digit only - the units digit. A number is even if it ends in 0, 2, 4, 6 or 8; even numbers are exactly divisible by 2 and can be written as 2n, so they pair off with nothing left over. A number is odd if it ends in 1, 3, 5, 7 or 9; divided by 2 an odd number leaves remainder 1 and can be written as 2n + 1, so one item is always unpaired. Note that zero is even (0 divided by 2 is 0, no remainder), and only the units digit counts - 3214 is even because it ends in 4, never mind the 3 at the front. The operation rules are heavily tested. Addition and subtraction: even and even give even, odd and odd give even, mixing odd and even gives odd. Multiplication: if any factor is even the product is even, and a product is odd only when every factor is odd (3 x 5 = 15, odd; 6 x 3 = 18, even). Two handy results: the sum of the first n odd numbers is n^2 (1 + 3 = 4 = 2^2; 1 + 3 + 5 + 7 = 16 = 4^2), and the sum of the first n even numbers is n(n + 1). The two errors children make most are assuming odd + odd stays odd and judging parity from the first digit rather than the last.

✅ Solved examples

1. Is 3214 odd or even? How do you know?
Even. Parity is decided by the units digit only, and 3214 ends in 4, which is even. The leading digits are irrelevant.
2. A child claims odd + odd = odd. Use 3 + 5 to correct this.
3 + 5 = 8, which is even. Algebraically (2m + 1) + (2n + 1) = 2(m + n + 1), always even. Odd + odd = even.
3. Without multiplying fully, state whether 6 x 7 is odd or even, and why.
Even. One factor (6) is even, and any product with an even factor is a multiple of 2. (6 x 7 = 42, even.)
4. Find the sum of the first four odd numbers and state the shortcut.
1 + 3 + 5 + 7 = 16 = 4^2. The sum of the first n odd numbers is n^2, so for n = 4 it is 16.

✏️ Practice — try these, take hints as needed

1. Is 0 odd or even?
0 divided by 2 leaves no remainder.
It fits 2n with n = 0.
Even
2. Even + Odd = ?
Try 8 + 7.
2m + (2n + 1) = 2(m+n) + 1.
Odd
3. Is the product 5 x 9 odd or even?
Both factors are odd.
Odd only when every factor is odd.
Odd (45)
4. The sum of the first five odd numbers (1 + 3 + 5 + 7 + 9) equals:
Use n^2.
n = 5.
25

📝 Topic test — 8 questions

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