What Makes a Number a Factor?
A factor slides into a number with nothing left over - no remainder, no fraction. Every whole number, no matter how small, has at least two built in: 1 and itself.
12 ÷ 1 = 12 ✓ → 1 is a factor
12 ÷ 2 = 6 ✓ → 2 is a factor
12 ÷ 3 = 4 ✓ → 3 is a factor
12 ÷ 4 = 3 ✓ → 4 is a factor
12 ÷ 5 = 2.4 ✗ → 5 is not a factor
12 ÷ 6 = 2 ✓ → 6 is a factor
12 ÷ 12 = 1 ✓ → 12 is a factor
Factors of 12: 1, 2, 3, 4, 6, 12
What Makes a Number a Multiple?
Multiply a number by any positive whole number and the result is one of its multiples. Since you can keep multiplying forever, every number has an endless list of them.
4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16...
Multiples of 4: 4, 8, 12, 16, 20, 24...
Factor Pairs
Line factors up in pairs and each pair multiplies back to the original number:
1 × 18, 2 × 9, 3 × 6
Factors: 1, 2, 3, 6, 9, 18
Common Factors
A common factor is one that fits evenly into two or more numbers at once:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Greatest Common Factor (GCF): 6
Common Multiples
A common multiple shows up in more than one number's times table:
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
Common multiples: 12, 24, 36...
Lowest Common Multiple (LCM): 12
Why This Distinction Actually Matters
| Application | Uses |
|---|---|
| Cutting down a fraction | Divide the top and bottom by their GCF to reach lowest terms |
| Combining fractions | Use the LCM of the denominators as a shared base |
| Prime factorization | Reduces a number to its prime building blocks |
| Divisibility checks | Tell instantly whether one number splits evenly into another |
Check Your Work With a Calculator
Working GCF and LCM out by hand is good practice, but for bigger numbers the SolverCalc GCF and LCM calculator, a quick calc solver built for exactly this kind of problem, finds both instantly and draws a Venn diagram of the shared prime factors so you can see where the overlap actually comes from.