Divisibility Rules and Factors
Grade 6 · Math · Study Guide
Learn quick mental math tricks to test whether one number divides evenly into another.
A number is divisible by another number if dividing produces a whole number with no remainder. Instead of doing long division every time, divisibility rules are shortcuts that let you test divisibility in your head. These are incredibly useful for simplifying fractions, finding GCF, and factoring.
For 2: A number is divisible by 2 if it ends in an even digit (0, 2, 4, 6, 8). For 3: Add all the digits — if the sum is divisible by 3, so is the number. For example, 372: 3+7+2=12, and 12÷3=4, so 372 is divisible by 3. For 5: ends in 0 or 5. For 10: ends in 0. For 9: digit sum divisible by 9. For 6: divisible by both 2 AND 3.
A factor of a number divides it evenly with no remainder. 1, 2, 3, 4, 6, and 12 are all factors of 12. Every number is divisible by 1 and itself. To list all factors, start at 1 and work up, looking for pairs: 1×12, 2×6, 3×4.
Divisibility rules are shortcuts for the brain that professional mathematicians actually use. They speed up prime factorization, simplifying fractions, and checking if large numbers divide evenly — without a calculator.
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0,2,4,6,8) | 348 → 8 is even ✓ |
| 3 | Sum of digits divisible by 3 | 123 → 1+2+3=6, 6÷3=2 ✓ |
| 5 | Last digit is 0 or 5 | 275 → ends in 5 ✓ |
| 6 | Divisible by both 2 AND 3 | 114 → even ✓, 1+1+4=6 ✓ |
| 9 | Sum of digits divisible by 9 | 513 → 5+1+3=9 ✓ |
| 10 | Last digit is 0 | 450 → ends in 0 ✓ |
Key terms
- Divisible
- A number is divisible by another if it divides evenly with no remainder
- Divisibility Rule
- A shortcut to check if one number divides evenly into another without doing long division
- Factor
- A whole number that divides evenly into another number
- Remainder
- The amount left over after division when a number doesn't divide evenly
Flashcards
- Divisibility rule for 2
- Last digit is even (0, 2, 4, 6, 8)
- Divisibility rule for 3
- Sum of all digits is divisible by 3
- Divisibility rule for 5
- Last digit is 0 or 5
- Divisibility rule for 9
- Sum of all digits is divisible by 9
- Divisibility rule for 6
- Number is divisible by BOTH 2 and 3
- How to list all factors
- Find factor pairs starting from 1: 1×n, 2×?, 3×?, up to the square root
Practice questions
Is 426 divisible by 3?
- Yes, because 4+2+6=12 and 12÷3=4
- No, because 426 is even
- Yes, because it ends in 6
- No, because 426 doesn't end in 3
Show answer
Answer: Yes, because 4+2+6=12 and 12÷3=4
The digit sum rule: 4+2+6=12, and 12 is divisible by 3, so 426 is divisible by 3.
Which of these numbers is divisible by both 2 and 5?
- 370
- 355
- 462
- 225
Show answer
Answer: 370
Divisible by 2: must end in even digit. Divisible by 5: must end in 0 or 5. Only 370 satisfies both (ends in 0).
Every number divisible by 6 must be divisible by both 2 and 3.
Show answer
Answer: True
6 = 2 × 3, so any multiple of 6 must include both 2 and 3 as factors.
To check divisibility by 9, add all the ___ and see if the sum is divisible by 9.
Show answer
Answer: digits
The divisibility rule for 9: add all the digits of the number. If the sum is divisible by 9, so is the original number.
How many factors does 18 have?
- 6
- 4
- 3
- 9
Show answer
Answer: 6
Factors of 18: 1, 2, 3, 6, 9, 18 — that's 6 factors. Pairs: 1×18, 2×9, 3×6.
Fun facts
- There is no simple divisibility rule for 7 — the closest shortcut requires several steps, which is why mathematicians say 7 is the 'awkward' prime.
- Any number that is divisible by both 2 and 3 is automatically divisible by 6 — you get two tests for the price of one!
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