Sums and differences
Whole tiles add to whole tiles. But leftovers can fit together too.
A number divisible by 8
72 is divisible by 8. It can be split as 48 + 24, and 8 divides both 48 and 24. But 72 can also be split as 50 + 22.
Does 8 divide 50 and 22?
What this lesson covers
The idea
If a divides M and a divides N, then a divides M + N and M – N; the converse need not hold, since a multiple of a can also be a sum of two non-multiples of a.
A number divisible by 8
72 is divisible by 8. It can be split as 48 + 24, and 8 divides both 48 and 24. But 72 can also be split as 50 + 22.
Does 8 divide 50 and 22?
- Yes, both
- Neither of them
- Only one of them
Join and take away
Each number is laid out in rows of 8. Change A and B, and switch between A + B and A − B. Look at the leftover cells.
Multiples add and subtract to multiples
If a divides M and a divides N, then a divides M + N and M – N; the converse need not hold, since a multiple of a can also be a sum of two non-multiples of a.
8a + 8b = 8(a + b) 8a − 8b = 8(a − b)
Examples: 8 + 16 = 24 16 + 56 = 72 80 + 120 = 200
The other way round fails: 72 = 50 + 22 is a multiple of 8, but 50 and 22 are not. Their leftovers 2 and 6 fit together to fill a row of 8.
Notes
If a divides M and a divides N, then a divides M + N and M – N; the converse need not hold, since a multiple of a can also be a sum of two non-multiples of a.
Check yourself
72 is divisible by 8. Which way of splitting 72 into two numbers does not have both parts divisible by 8?
M leaves remainder 6 and N leaves remainder 5 when divided by 8. What remainder does M + N leave?
Answer: 3
6 + 5 = 11 leftover cells. 8 of them fill a row, so 11 − 8 = 3 are left over. The remainder is 3.
Which statement is always true?
M = 56 and N = 24 are both multiples of 8. What is (M − N) ÷ 8?
Answer: 4
M − N = 56 − 24 = 32 and 32 ÷ 8 = 4. Or: 8 × 7 − 8 × 3 = 8 × (7 − 3) = 8 × 4.
- 48 + 24. Both 48 = 8 × 6 and 24 = 8 × 3 are multiples of 8.
- 40 + 32. Both 40 = 8 × 5 and 32 = 8 × 4 are multiples of 8.
- 50 + 22 — correct. Yes! 50 leaves 2 and 22 leaves 6. The leftovers fit together, but neither number is a multiple of 8.
- 64 + 8. Both 64 = 8 × 8 and 8 = 8 × 1 are multiples of 8.
- If 6 divides a number, then 6 divides any two numbers that add up to it. 14 + 16 = 30 and 6 divides 30, but 6 divides neither 14 nor 16.
- If 6 divides two numbers, then 6 divides their sum — correct. Yes! 6a + 6b = 6(a + b), so 6 is a factor of the sum.
- If 6 divides the sum of two numbers, then 6 divides each of them. 25 + 35 = 60 and 6 divides 60, but 6 divides neither 25 nor 35.