Divisibility Laws
Understanding the Mathematical Rules of Division
1What is Divisibility?
Definition
A number a is divisible by number b if when we divide a ÷ b, we get a whole number with no remainder.
We say: "b divides a" or "a is a multiple of b"
✓ Examples of Divisibility:
- 24 ÷ 6 = 4 (exact!) → 24 is divisible by 6
- 100,000 ÷ 8 = 12,500 (exact!) → 100,000 is divisible by 8
- 45,000 ÷ 9 = 5,000 (exact!) → 45,000 is divisible by 9
24 items divided into 4 equal groups of 6 — no leftovers!
2Notation for Divisibility
Mathematical Symbols
| Symbol | Meaning | Example |
|---|---|---|
| a ⋮ b | "a is divisible by b" | 24 ⋮ 6 |
| b | a | "b divides a" | 6 | 24 |
When a Number is NOT Divisible
When division produces a remainder, the number is NOT divisible.
- 25 ÷ 6 = 4 R1 → 25 is NOT divisible by 6
- 100,000 ÷ 7 = 14,285 R5 → 100,000 is NOT divisible by 7
3LAW 1: Divisibility of a Sum
Formal Statement
If each of two natural numbers is divisible by some natural number, then their sum is also divisible by that number, and the resulting quotient equals the sum of the quotients obtained by dividing each addend.
In Symbols:
If a ⋮ c and b ⋮ c, then (a + b) ⋮ c
Moreover: (a + b) ÷ c = (a ÷ c) + (b ÷ c)
If a ⋮ c and b ⋮ c, then (a + b) ⋮ c
Moreover: (a + b) ÷ c = (a ÷ c) + (b ÷ c)
Algebraically:
If a = c × m and b = c × n
Then a + b = c × m + c × n = c × (m + n)
If a = c × m and b = c × n
Then a + b = c × m + c × n = c × (m + n)
Proof of Law 1
- Let a ÷ c = m and b ÷ c = n (where m and n are whole numbers)
- Then a = c × m and b = c × n
- So a + b = c × m + c × n = c × (m + n)
- Therefore (a + b) ÷ c = m + n = (a ÷ c) + (b ÷ c) ✓