Divisibility
Rules
Master the quick shortcuts that let you test whether one number divides evenly into another: no long division required.
SSAT Middle Level
What Does Divisible Mean?
A number \(a\) is divisible by another number \(b\) when the division \(\frac{a}{b}\) produces a whole number with no remainder.
Factors, Multiples & Divisibility
Factor
A number that divides evenly into another. \(4\) is a factor of \(20\).
Multiple
The product you get when you multiply by a whole number. \(20\) is a multiple of \(4\).
Divisibility
The relationship linking both: if \(b\) is a factor of \(a\), then \(a\) is divisible by \(b\).
Why Learn Divisibility Rules?
1
Speed: Test factors in seconds without performing full division.
2
Simplify fractions: Find the GCF quickly to reduce \(\frac{36}{48}\) to \(\frac{3}{4}\).
3
Prime factorization: Break any number into primes by testing small divisors first.
4
SSAT strategy: Eliminate answer choices and spot multiples in word problems.
Rules for 2, 5, and 10
These three rules all depend on the last digit of the number.
Divisible by 2
Last digit is even: \(0, 2, 4, 6, 8\).
Example: \(1{,}374\) ends in \(4\) ✓
Example: \(1{,}374\) ends in \(4\) ✓
Divisible by 5
Last digit is \(0\) or \(5\).
Example: \(2{,}085\) ends in \(5\) ✓
Example: \(2{,}085\) ends in \(5\) ✓
Divisible by 10
Last digit is \(0\).
Example: \(5{,}430\) ends in \(0\) ✓
Example: \(5{,}430\) ends in \(0\) ✓
Rules for 3 and 9
These rules use the digit sum: add up all the digits of the number.
Divisible by 3
The sum of the digits is divisible by \(3\).
Example: \(528 \to 5+2+8=15\), and \(15 \div 3 = 5\) ✓
Example: \(528 \to 5+2+8=15\), and \(15 \div 3 = 5\) ✓
Divisible by 9
The sum of the digits is divisible by \(9\).
Example: \(738 \to 7+3+8=18\), and \(18 \div 9 = 2\) ✓
Example: \(738 \to 7+3+8=18\), and \(18 \div 9 = 2\) ✓
Rules for 4 and 6
Divisible by 4
The last two digits form a number divisible by \(4\).
Example: \(1{,}316 \to 16 \div 4 = 4\) ✓
Example: \(1{,}316 \to 16 \div 4 = 4\) ✓
Divisible by 6
The number is divisible by both \(2\) and \(3\).
Example: \(534\): even ✓ and \(5+3+4=12\), divisible by \(3\) ✓
Example: \(534\): even ✓ and \(5+3+4=12\), divisible by \(3\) ✓
Rules for 8 and 11
Divisible by 8
The last three digits form a number divisible by \(8\).
Example: \(5{,}320 \to 320 \div 8 = 40\) ✓
Example: \(5{,}320 \to 320 \div 8 = 40\) ✓
Divisible by 11
Alternating sum of digits is divisible by \(11\) (or equals \(0\)).
Example: \(9{,}174 \to 9-1+7-4=11\) ✓
Example: \(9{,}174 \to 9-1+7-4=11\) ✓
3 vs 9: What's the Difference?
Both rules use the digit sum, but the threshold is different. Many students confuse them.
| Number | Digit Sum | Div by 3? | Div by 9? |
|---|---|---|---|
| \(123\) | \(6\) | ✓ Yes | ✕ No |
| \(279\) | \(18\) | ✓ Yes | ✓ Yes |
| \(145\) | \(10\) | ✕ No | ✕ No |
| \(981\) | \(18\) | ✓ Yes | ✓ Yes |
Visualising the Digit Sum Rule
How to Apply the Rules
1
Read the divisor. Which number are you testing: \(2\), \(3\), \(4\), \(5\), \(6\), \(8\), \(9\), \(10\), or \(11\)?
2
Pick the matching rule. Last digit? Digit sum? Last two/three digits? Alternating sum? Combination test?
3
Extract the piece. Write down only the digits or sum you need.
4
Test the piece. Is it divisible by the target? If yes, the whole number is divisible.
5
State your answer. Confirm "divisible" or "not divisible" with a brief reason.
Example 1: Is \(2{,}346\) Divisible by \(6\)?
1
Test for 2: The last digit is \(6\), which is even. ✓ Divisible by \(2\).
2
Test for 3: Digit sum = \(2+3+4+6 =\) \(15\). Is \(15 \div 3 = 5\)? ✓ Divisible by \(3\).
3
Conclusion: Since \(2{,}346\) passes both tests, it is divisible by \(6\).
Example 2: Is \(7{,}524\) Divisible by \(4\)? By \(8\)?
1
Test for 4: Look at the last two digits: \(24\). Is \(24 \div 4 = 6\)? ✓ Divisible by 4.
2
Test for 8: Look at the last three digits: \(524\). Is \(524 \div 8 = 65.5\)? ✕ Not divisible by 8.
Example 3: Is \(85{,}294\) Divisible by \(11\)?
Common Mistakes
✕ Confusing 3 and 9
Digit sum divisible by \(3\) does NOT guarantee divisible by \(9\). Always check the correct divisor.
✕ Wrong digits for 4 vs 8
Rule of \(4\) uses the last two digits. Rule of \(8\) uses the last three. Mixing them up gives wrong answers.
✕ Testing only one part of 6
A number must pass both the test for \(2\) AND the test for \(3\) to be divisible by \(6\). Checking only one is insufficient.
Complete Reference Table
| Divisor | Rule | Quick Check |
|---|---|---|
| \(2\) | Last digit is even | Look at ones place |
| \(3\) | Digit sum divisible by \(3\) | Add all digits |
| \(4\) | Last 2 digits divisible by \(4\) | Check two-digit number |
| \(5\) | Last digit is \(0\) or \(5\) | Look at ones place |
| \(6\) | Divisible by \(2\) AND \(3\) | Two separate tests |
| \(8\) | Last 3 digits divisible by \(8\) | Check three-digit number |
| \(9\) | Digit sum divisible by \(9\) | Add all digits |
| \(10\) | Last digit is \(0\) | Look at ones place |
| \(11\) | Alternating sum divisible by \(11\) | \(+, -, +, -, \dots\) |
Question 1 of 5
Which of the following numbers is divisible by \(9\)?
The digit sum of \(6{,}318\) is \(6+3+1+8 = 18\), and \(18 \div 9 = 2\). The other digit sums (\(14\), \(19\), \(14\)) are not divisible by \(9\).
Question 2 of 5
A number is divisible by \(6\). Which of the following must also be true?
Since \(6 = 2 \times 3\), any multiple of \(6\) is automatically a multiple of both \(2\) and \(3\). The other options are not guaranteed: for example, \(18\) is divisible by \(6\) but not by \(4\) or \(12\).
Question 3 of 5
Is \(3{,}716\) divisible by \(4\)?
The rule for \(4\) checks the last two digits. The last two digits are \(16\), and \(16 \div 4 = 4\) exactly. Being even (choice C) only proves divisibility by \(2\), not \(4\). Choice B wrongly uses the digit-sum rule, which applies to \(3\) and \(9\).
Question 4 of 5
What is the alternating sum of the digits of \(5{,}907\), and is it divisible by \(11\)?
Alternating sum: \(5 - 9 + 0 - 7 = -11\). Since \(-11\) is a multiple of \(11\), the number is divisible by \(11\). Negative multiples count too.
Question 5 of 5
Which number is divisible by both \(3\) and \(5\) but not by \(9\)?
\(735\) ends in \(5\) (divisible by \(5\) ✓), its digit sum is \(7+3+5=15\) (divisible by \(3\) ✓), but \(15\) is NOT divisible by \(9\) (✕). The others: \(945\) has a digit sum of \(18\) (divisible by \(9\)), \(810\) has a digit sum of \(9\) (divisible by \(9\)), and \(450\) has a digit sum of \(9\) (divisible by \(9\)).