Two kinds of even numbers
Rows of four: even numbers leave 0 or 2.
Even numbers in rows of four
Put dots in rows of 4. 8 dots make two full rows. 10 dots make two full rows and a short row. Now take any even number of dots.
How many dots can be left in the short last row?
What this lesson covers
The idea
On division by 4, an even number leaves remainder 0 if it is a multiple of 4 (of the form 4p) and remainder 2 if it is not (of the form 4p + 2).
Even numbers in rows of four
Put dots in rows of 4. 8 dots make two full rows. 10 dots make two full rows and a short row. Now take any even number of dots.
How many dots can be left in the short last row?
- Only 0 or 2
- Any of 0, 1, 2, 3
- Always 2
Sort the even numbers
Step through the even numbers. Each one is laid out in rows of 4. Watch the cells left over, and see which column the number drops into.
4p or 4p + 2
On division by 4, an even number leaves remainder 0 if it is a multiple of 4 (of the form 4p) and remainder 2 if it is not (of the form 4p + 2).
4p is p full rows of 4. 4p + 2 is p full rows and 2 more cells. Every even number is one of these two kinds. Remainder 1 or 3 is impossible: whole rows are even, so the cells left over must be even too.
Examples: 12 = 4 × 3 14 = 4 × 3 + 2 38 = 4 × 9 + 2
Notes
On division by 4, an even number leaves remainder 0 if it is a multiple of 4 (of the form 4p) and remainder 2 if it is not (of the form 4p + 2).
Check yourself
Which of these is an even number that is not a multiple of 4?
Write 38 in the form 4p + 2. What is p?
Answer: 9
38 = 36 + 2 and 36 = 4 × 9, so 38 = 4 × 9 + 2 and p = 9.
How many of the even numbers 6, 12, 18, 24, 30, 36 are of the form 4p + 2?
Answer: 3
6 = 4 × 1 + 2, 18 = 4 × 4 + 2 and 30 = 4 × 7 + 2 leave remainder 2. 12, 24 and 36 are multiples of 4. So 3 numbers are of the form 4p + 2.
Why can an even number never leave remainder 1 or 3 when divided by 4?
- 16. 16 = 4 × 4 is a multiple of 4.
- 20. 20 = 4 × 5 is a multiple of 4.
- 26 — correct. Yes! 26 = 4 × 6 + 2. It leaves remainder 2.
- 32. 32 = 4 × 8 is a multiple of 4.
- Only multiples of 4 are even. 6 is even but it is not a multiple of 4 (6 = 4 + 2).
- Whole rows of 4 are even, so the cells left over must be even: 0 or 2 — correct. Yes! Even total, even rows: the leftover cells must be even. With fewer than 4 cells left, that means 0 or 2.
- The remainder is always 2. The remainder is 0 for 4, 8, 12 … so it is not always 2.