Divisible and Factors- Practice Problems and Rules

Divisibility Rules: What Actually Works

Math class throws a lot at you. Divisibility rules are supposed to make your life easier, but half the time you forget them mid-exam. This guide cuts through the noise. You'll learn the rules that matter, how factors actually work, and you'll get practice problems with real solutions.

No motivational nonsense. Just the math.

What Divisibility Actually Means

A number is divisible by another number if you can divide it evenly, with no remainder. That's it. When you divide 12 by 3, you get 4 with no leftover. So 12 is divisible by 3.

When there's a remainder, it's not divisible. 10 divided by 4 gives you 2 with a remainder of 2. So 10 is not divisible by 4.

The Divisibility Rules That Save Time

Divisibility by 2

If the last digit is even (0, 2, 4, 6, 8), the number is divisible by 2.

Examples: 346 ✓ | 7,892 ✓ | 531 ✗

Divisibility by 3

Add up all the digits. If that sum is divisible by 3, the original number is too.

Example: 4,716 → 4+7+1+6 = 18 → 18 is divisible by 3 → 4,716 is divisible by 3

Divisibility by 4

Check the last two digits. If those two digits form a number divisible by 4, the whole thing is divisible by 4.

Example: 1,248 → last two digits are 48 → 48 ÷ 4 = 12 → 1,248 is divisible by 4

Divisibility by 5

Last digit is 0 or 5. That's the whole rule.

Examples: 1,230 ✓ | 895 ✓ | 4,001 ✗

Divisibility by 6

The number must be divisible by BOTH 2 and 3. If it passes both tests, it's divisible by 6.

Example: 114 → even (passes 2) → 1+1+4 = 6 (passes 3) → 114 is divisible by 6

Divisibility by 7

Take the last digit, double it, and subtract from the rest of the number. If that result is divisible by 7, so is the original.

Example: 203 → last digit is 3 → 3 × 2 = 6 → 20 - 6 = 14 → 14 is divisible by 7 → 203 is divisible by 7

Divisibility by 8

Check the last three digits. If that three-digit number is divisible by 8, so is the whole thing.

Example: 5,832 → last three digits are 832 → 832 ÷ 8 = 104 → 5,832 is divisible by 8

Divisibility by 9

Same deal as 3. Add all digits. If the sum is divisible by 9, the original number is too.

Example: 2,457 → 2+4+5+7 = 18 → 18 is divisible by 9 → 2,457 is divisible by 9

Divisibility by 10

Last digit must be 0. That's it.

Divisibility by 11

Take alternating digits and find the difference. If that difference is divisible by 11 (or equals 0), the number is too.

Example: 4,618 → (4+8) - (6+1) = 12 - 7 = 5 → not divisible by 11 → 4,618 is not divisible by 11

Example: 3,437 → (3+3) - (4+7) = 6 - 11 = -5 → not divisible by 11 → wait, let me check 121 instead...

Example: 121 → (1+1) - 2 = 0 → 121 is divisible by 11 ✓

Divisibility by 12

The number must be divisible by BOTH 3 and 4. That's the easiest way to check.

Example: 144 → 1+4+4 = 9 (passes 3) → last two digits 44 ÷ 4 = 11 (passes 4) → 144 is divisible by 12

Quick Reference: Divisibility Rules Comparison

Divisor Rule Example
2 Last digit is even 4,892 ✓
3 Sum of digits divisible by 3 1,254 → 12 ✓
4 Last two digits divisible by 4 716 → 16 ✓
5 Last digit is 0 or 5 3,045 ✓
6 Divisible by 2 AND 3 102 ✓
7 Double last digit, subtract from rest 203 → 20-6=14 ✓
8 Last three digits divisible by 8 3,000 ✓
9 Sum of digits divisible by 9 4,221 → 9 ✓
10 Last digit is 0 890 ✓
11 Alternating digit difference divisible by 11 121 → 0 ✓
12 Divisible by 3 AND 4 144 ✓

What Are Factors?

Factors are numbers that divide evenly into another number. The factors of 12 are 1, 2, 3, 4, 6, and 12. Each of these divides 12 without leaving a remainder.

Every number has at least two factors: 1 and itself. Prime numbers only have those two. Composite numbers have more.

Prime vs Composite Numbers

How to Find All Factors of a Number

Here's the straightforward method:

  1. Start with 1 and your number
  2. Test each integer from 2 upward
  3. When you hit a pair where both numbers work, write them down
  4. Stop when your test number exceeds the square root of your target

Why stop at the square root? Because factor pairs multiply to the original number. Once you pass the square root, you're just repeating pairs in reverse.

Example: Finding Factors of 36

Square root of 36 is 6. Test numbers 1 through 6:

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Practice Problems

Test yourself. Answers below.

Problem Set 1: Apply the Divisibility Rules

  1. Is 5,184 divisible by 3?
  2. Is 7,290 divisible by 4?
  3. Is 1,001 divisible by 11?
  4. Is 144 divisible by 6?
  5. Is 9,999 divisible by 9?

Problem Set 2: Find All Factors

  1. Find all factors of 24
  2. Find all factors of 45
  3. Find all factors of 17

Problem Set 3: Identify Prime or Composite

  1. Is 27 prime or composite?
  2. Is 31 prime or composite?
  3. Is 49 prime or composite?

Answers and Solutions

Problem Set 1 Solutions

  1. 5,184: 5+1+8+4 = 18 → 18 ÷ 3 = 6 ✓ Yes
  2. 7,290: last two digits are 90 → 90 ÷ 4 = 22.5 ✗ No
  3. 1,001: (1+0) - (0+1) = 0 → 0 is divisible by 11 ✓ Yes
  4. 144: even ✓, 1+4+4 = 9 (divisible by 3) ✓ → Yes
  5. 9,999: 9+9+9+9 = 36 → 36 ÷ 9 = 4 ✓ Yes

Problem Set 2 Solutions

  1. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  2. Factors of 45: 1, 3, 5, 9, 15, 45
  3. Factors of 17: 1, 17 (17 is prime)

Problem Set 3 Solutions

  1. 27 is composite (factors: 1, 3, 9, 27)
  2. 31 is prime (only 1 and 31 divide evenly)
  3. 49 is composite (7 × 7 = 49)

Getting Started: Your Action Plan

To get fast at this:

  1. Memorize the easy rules first: 2, 5, 10. They take 30 seconds to learn.
  2. Learn 3 and 9 together. Same process, just different divisors. Sum the digits.
  3. Practice finding factors with the square root method. Start with numbers under 100.
  4. Test yourself daily. Pick a random number and list its factors. Time yourself.
  5. Know your primes up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Common Mistakes to Avoid

That's the full picture. Memorize the rules that come up constantly (2, 3, 4, 5, 9, 10), and you'll handle most problems without breaking a sweat.