Q1.
1. If 8 exactly divides two numbers separately, it must exactly divide their sum. Statement 1 is always true. Determine if it is true with subtraction.
Answer
Always true for addition, and always true for subtraction as well.
The two numbers are multiples of 8: 8a and 8b
8a + 8b = 8(a + b) → a multiple of 8
8a – 8b = 8(a – b) → also a multiple of 8
8a + 8b = 8(a + b) → a multiple of 8
8a – 8b = 8(a – b) → also a multiple of 8
| Two multiples of 8 | Sum | Difference |
|---|---|---|
| 8 and 16 | 24 = 8 × 3 | – 8 = 8 × (– 1) |
| 16 and 56 | 72 = 8 × 9 | – 40 = 8 × (– 5) |
| 80 and 120 | 200 = 8 × 25 | – 40 = 8 × (– 5) |
Why it happens: multiples of 8 are what you get by adding 8 over and over. Adding two such piles, or taking one away from the other, still leaves a whole number of 8s — no partial group is ever created. In symbols, 8 comes out as a common factor either way.
Tip: this is the general fact if a | M and a | N, then a | M + N and a | M – N. Nothing about 8 was special.