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Factors, Multiples, Primes, GCF and LCM

Find factors, multiples, primes and prime factorizations, and use the GCF and LCM to solve word problems.

Practice this skill 22 min Multiples and factors Leer en español
Key idea

What you'll learn

This lesson covers the "building blocks" of whole numbers: factors, multiples, prime numbers, divisibility rules, the greatest common factor (GCF) and the least common multiple (LCM).

How it shows up on the GED:

  • Direct questions: "What is the GCF of 42 and 56?", "Which number is prime?", "Which is the prime factorization of 180?"
  • Word problems that hide GCF or LCM: splitting items into identical groups, buying packages so nothing is left over, schedules that line up again.
  • Algebra: factoring out a GCF, like 12x + 18 = 6(2x + 3), and finding common denominators for fractions (the LCM of the denominators).

In real life you use these ideas when you share things evenly, plan schedules, cut material into equal pieces, or add fractions in a recipe. Most of this skill is tested without a calculator, so the mental shortcuts in this lesson really pay off.

Words to know

  • Factor — a whole number that divides another with no remainder (4 is a factor of 12 because 12 \div 4 = 3).
  • Factor pair — two factors that multiply to the number (3 \times 4 = 12).
  • Multiple — the result of multiplying a number by 1, 2, 3, \ldots (multiples of 5: 5, 10, 15, \ldots).
  • Divisible — divides evenly, with remainder 0 (35 is divisible by 7).
  • Prime number — a whole number greater than 1 with exactly two factors, 1 and itself (2, 3, 5, 7, 11, 13).
  • Composite number — a whole number greater than 1 with more than two factors (9 = 3 \times 3).
  • Prime factorization — writing a number as a product of primes only (60 = 2^2 \times 3 \times 5).
  • Factor tree — a diagram that splits a number into factors until all branches end in primes.
  • Common factor — a factor shared by two numbers (3 is a common factor of 12 and 15).
  • Greatest common factor (GCF) — the largest common factor (\text{GCF}(12, 15) = 3).
  • Common multiple — a multiple shared by two numbers (30 is a common multiple of 6 and 10).
  • Least common multiple (LCM) — the smallest positive common multiple (\text{LCM}(6, 10) = 30).
  • Exponent — a small raised number that counts repeated factors (2^3 = 2 \times 2 \times 2).

All Math words in the glossary

Factors and multiples: two sides of the same fact

The fact 3 \times 4 = 12 tells us two things:

  • 3 and 4 are factors of 12 (they go into 12).
  • 12 is a multiple of 3 and of 4 (it is reached by counting by 3s or 4s).

Finding all factors — use factor pairs. Start at 1 and work upward; stop when the pairs meet.

Example: factors of 36: 1 \times 36, 2 \times 18, 3 \times 12, 4 \times 9, 6 \times 6. So the factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Factor rainbow for 36. The factors 1, 2, 3, 4, 6, 9, 12, 18, 36 are written in order. Arcs join each factor pair: 1 × 36, 2 × 18, 3 × 12 and 4 × 9; the middle factor 6 pairs with itself, 6 × 6. 1 × 36 2 × 18 3 × 12 4 × 9 1 2 3 4 6 9 12 18 36 6 × 6

Listing multiples — count by the number: multiples of 7 are 7, 14, 21, 28, 35, \ldots The list never ends.

Key facts:

  • Every number is a factor and a multiple of itself.
  • 1 is a factor of every number.
  • A factor is never larger than the number; a multiple is never smaller (for positive numbers).

Prime vs composite. A prime has exactly two factors. The primes under 50 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Grid of the numbers 1 to 50 in rows of ten. The 15 primes are shaded blue: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. The number 1 is outlined with a dashed box because it is neither prime nor composite. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Blue = prime (15 primes under 50). 2 is the only even prime. 1 (dashed) is neither prime nor composite.

Notes: 1 is neither prime nor composite. 2 is the only even prime. Many odd numbers are not prime: 9, 15, 21, 27, 33, 39, 49, 51, 57, 91.

To test whether a number n is prime, try dividing by the primes 2, 3, 5, 7, 11, \ldots — you can stop once the prime squared is bigger than n. For 97: try 2, 3, 5, 7 (since 11^2 = 121 > 97). None divide, so 97 is prime.

Divisibility rules and prime factorization

Divisibility rules let you test factors quickly without a calculator.

Divisible byRuleExample
2last digit is even (0, 2, 4, 6, 8)358 ✓
3sum of digits divisible by 3471: 4+7+1=12 ✓
4last two digits divisible by 41{,}316: 16 ✓
5last digit 0 or 5745 ✓
6divisible by 2 and 3522: even, 5+2+2=9 ✓
8last three digits divisible by 85{,}120: 120 \div 8 = 15 ✓
9sum of digits divisible by 98{,}433: 18 ✓
10last digit 0930 ✓
25last two digits 00, 25, 50 or 751{,}475 ✓

Prime factorization with a factor tree. Split the number into any two factors, then split again until every branch is prime. Circle the primes and multiply them back as a check. Whichever split you start with, you always reach the same primes.

Example: 360 = 36 \times 10 = (6 \times 6) \times (2 \times 5) = 2 \times 3 \times 2 \times 3 \times 2 \times 5 = 2^3 \times 3^2 \times 5.

Factor tree for 360. 360 splits into 36 and 10. 36 splits into 6 and 6; each 6 splits into 2 and 3. 10 splits into 2 and 5. The circled prime leaves are 2, 3, 2, 3, 2 and 5, so 360 = 2 x 2 x 2 x 3 x 3 x 5 = 2^3 x 3^2 x 5. 360 36 10 6 6 2 5 2 3 2 3 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5

Factor tree for 360. Every branch ends in a circled prime: 360 = 2^3 \times 3^2 \times 5.

GCF and LCM: how to find them and when to use them

GCF — Method 1 (list): list the factors of each number and pick the biggest shared one. \text{GCF}(18, 30): factors of 18: 1, 2, 3, 6, 9, 18; of 30: 1, 2, 3, 5, 6, 10, 15, 30. GCF = 6.

GCF — Method 2 (primes): take the primes the numbers share, each with the smaller power. 18 = 2 \times 3^2, 30 = 2 \times 3 \times 5 → GCF = 2 \times 3 = 6.

Venn diagram of prime factors. Left circle: 18 = 2 × 3 × 3. Right circle: 30 = 2 × 3 × 5. The overlap holds the shared primes 2 and 3; 3 is only in the 18 circle and 5 only in the 30 circle. GCF = 2 × 3 = 6. 18 = 2 × 3 × 3 30 = 2 × 3 × 5 3 2 3 5 GCF = product of the overlap = 2 × 3 = 6

LCM — Method 1 (list): list multiples of the larger number until one is divisible by the smaller. \text{LCM}(6, 9): 9, 18 — and 18 \div 6 = 3. LCM = 18.

LCM — Method 2 (primes): take every prime that appears, each with the larger power. 6 = 2 \times 3, 9 = 3^2 → LCM = 2 \times 3^2 = 18.

Two number lines from 0 to 36. Top: hops of 6 land on 6, 12, 18, 24, 30, 36. Bottom: hops of 9 land on 9, 18, 27, 36. A dashed green line marks 18, the first number both land on, so LCM(6, 9) = 18. Multiples of 6 0 6 12 18 24 30 36 Multiples of 9 0 9 18 27 36 first common multiple: LCM(6, 9) = 18

If two numbers share no factor except 1 (like 8 and 15), their GCF is 1 and their LCM is their product, 120.

Which one does the word problem want?

Clue wordsUseWhy
split into identical groups, greatest number of groups, largest equal pieces, biggest square tilesGCFthe answer divides into the given numbers, so it is small
same number of each, next time together, buy packages with none left over, repeat at the same timeLCMthe answer is something both numbers divide into, so it is large

Sanity check: a GCF is at most the smaller number; an LCM is at least the larger number.

Factoring out a GCF (algebra link). To factor 12x + 18, find \text{GCF}(12, 18) = 6 and write 6(2x + 3). Check by distributing: 6 \cdot 2x + 6 \cdot 3 = 12x + 18 ✓. With variables, \text{GCF}(8x^2, 12x) = 4x, so 8x^2 + 12x = 4x(2x + 3).

Area model for factoring 12x + 18. A rectangle has height 6. Its top is split into widths 2x and 3. The left part has area 6 × 2x = 12x and the right part has area 6 × 3 = 18. So 12x + 18 = 6(2x + 3). 2x 3 6 12x 18 12x + 18 = 6(2x + 3), since GCF(12, 18) = 6
Formula

Key relationship (not on the formula sheet)

None of the following is on the GED formula sheet.

For any two positive whole numbers a and b:

\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b
  • a, b = the two numbers
  • GCF = greatest common factor, LCM = least common multiple

Example: a = 12, b = 18: \text{GCF} = 6, \text{LCM} = 36, and 6 \times 36 = 216 = 12 \times 18 ✓.

This rule is handy in "find the missing number" problems. It works for two numbers only, not three.

Worked example

Example 1 (easy): GCF of 42 and 56

Find the GCF of 42 and 56.

  1. Prime factorizations: 42 = 2 \times 3 \times 7 and 56 = 2^3 \times 7.
  2. Shared primes: 2 and 7. Smaller powers: 2^1 and 7^1.
  3. GCF = 2 \times 7 = 14.
Venn diagram of prime factors. Left circle: 42 = 2 × 3 × 7. Right circle: 56 = 2 × 2 × 2 × 7. The overlap holds the shared primes 2 and 7; 3 is only in the 42 circle; 2 and 2 are only in the 56 circle. GCF = 2 × 7 = 14. 42 = 2 × 3 × 7 56 = 2 × 2 × 2 × 7 3 2 7 2 2 GCF = 2 × 7 = 14

Answer: 14.

Check: 42 \div 14 = 3 and 56 \div 14 = 4; 3 and 4 share no factor except 1. ✓

Worked example

Example 2: LCM of 12 and 15

Find the LCM of 12 and 15.

  1. Prime factorizations: 12 = 2^2 \times 3 and 15 = 3 \times 5.
  2. Every prime, with the larger power: 2^2, 3, 5.
  3. LCM = 4 \times 3 \times 5 = 60.
Venn diagram of prime factors. Left circle: 12 = 2 × 2 × 3. Right circle: 15 = 3 × 5. The overlap holds the shared prime 3; 2 and 2 are only in the 12 circle and 5 is only in the 15 circle. The LCM uses every prime in the diagram: 2 × 2 × 3 × 5 = 60. 12 = 2 × 2 × 3 15 = 3 × 5 2 2 3 5 LCM = every prime in the diagram = 2 × 2 × 3 × 5 = 60

Answer: 60.

Check: 60 \div 12 = 5 and 60 \div 15 = 4. And \text{GCF} = 3, so 3 \times 60 = 180 = 12 \times 15. ✓

Worked example

Example 3 (word problem, GCF): bake-sale trays

Keisha baked 36 cookies and 54 brownies. She wants to make identical trays using all of them. What is the greatest number of trays, and what goes on each tray?

  1. Identical groups using everything → GCF.
  2. 36 = 2^2 \times 3^2 and 54 = 2 \times 3^3. GCF = 2 \times 3^2 = 18.
  3. Per tray: 36 \div 18 = 2 cookies and 54 \div 18 = 3 brownies.
Eighteen identical trays arranged in 3 rows of 6. Each tray holds 2 cookies (blue circles) and 3 brownies (red squares). 18 × 2 = 36 cookies and 18 × 3 = 54 brownies. cookie brownie 18 trays: 18 × 2 = 36 cookies, 18 × 3 = 54 brownies

Answer: 18 trays, each with 2 cookies and 3 brownies.

Check: 18 \times 2 = 36 and 18 \times 3 = 54. ✓

Worked example

Example 4 (word problem, LCM): runners on a track

Marcus runs one lap every 6 minutes and Dana every 8 minutes. They start together at the starting line. After how many minutes will they first be at the starting line together again?

  1. "Together again" → LCM.
  2. Multiples of 8: 8, 16, 24. Is 24 divisible by 6? Yes (24 \div 6 = 4).
  3. LCM = 24.
Two timelines from 0 to 48 minutes. Top: Marcus finishes a lap every 6 minutes, at 6, 12, 18, 24, 30, 36, 42, 48. Bottom: Dana finishes a lap every 8 minutes, at 8, 16, 24, 32, 40, 48. A dashed green line marks 24 minutes, the first time both are at the starting line together. Marcus: every 6 min 0 6 12 18 24 30 36 42 48 Dana: every 8 min 0 8 16 24 32 40 48 together again after 24 min = LCM(6, 8)

Answer: after 24 minutes (Marcus has run 4 laps, Dana 3).

Check: 6 \times 4 = 24 and 8 \times 3 = 24. ✓

Worked example

Example 5 (GED-hard, work backwards): find the missing number

The GCF of two numbers is 4, and their LCM is 120. One number is 24. Find the other number.

  1. Use \text{GCF} \times \text{LCM} = a \times b: 4 \times 120 = 24 \times b.
  2. 480 = 24b, so b = 480 \div 24 = 20.
  3. Verify with primes: 24 = 2^3 \times 3, 20 = 2^2 \times 5.
    • GCF = 2^2 = 4 ✓
    • LCM = 2^3 \times 3 \times 5 = 120 ✓
Venn diagram of prime factors. Left circle: 24 = 2 × 2 × 2 × 3. Right circle: 20 = 2 × 2 × 5. The overlap holds the shared primes 2 and 2; 2 and 3 are only in the 24 circle and 5 is only in the 20 circle. GCF = 2 × 2 = 4 and LCM = 2 × 3 × 2 × 2 × 5 = 120. 24 = 2 × 2 × 2 × 3 20 = 2 × 2 × 5 2 3 2 2 5 GCF = overlap = 2 × 2 = 4 LCM = all primes = 2 × 3 × 2 × 2 × 5 = 120

Answer: 20.

Check: 120 \div 24 = 5 and 120 \div 20 = 6, and no smaller common multiple exists. ✓

Calculator tip

TI-30XS MultiView tips

Most factor and multiple questions are on the calculator-free part of the GED, so practice the divisibility rules and factor trees by hand.

When a calculator is allowed:

  • Test a factor: divide and look for a whole-number answer. 2340 ÷ 8 enter shows 292.5 — not whole, so 8 is not a factor.
  • Test for prime: divide by 3, 7, 11, 13, \ldots and look for a whole-number result.
  • Check a factorization: multiply it back out; use the ^ key for powers, e.g. 2 ^ 3 gives 8.
  • The n/d key can simplify a fraction, which reveals the GCF: entering \tfrac{48}{60} and pressing enter gives \tfrac{4}{5}, so the GCF is 48 \div 4 = 12.

When not to use it: listing multiples of small numbers is faster in your head than typing.

Common mistake

Common mistakes

  • Mixing up GCF and LCM. Wrong: "GCF of 6 and 8 is 24." Right: the GCF divides into both (2); the LCM is divisible by both (24).
  • Assuming the product is the LCM. Wrong: \text{LCM}(6, 8) = 48. Right: 48 is a common multiple, but 24 is the least. The product works only if the GCF is 1.
  • Thinking odd means prime. Wrong: "51 is prime." Right: 51 = 3 \times 17. Test small primes.
  • Calling 1 prime, or 2 composite. 1 is neither; 2 is prime (the only even prime).
  • Stopping the factor tree too early. Wrong: 180 = 4 \times 5 \times 9. Right: 4 and 9 are not prime; 180 = 2^2 \times 3^2 \times 5.
  • Answering the wrong quantity. In "how many bouquets?", the GCF is the answer, not the number of roses per bouquet.
Test strategy

GED test strategy

  • Decide GCF or LCM first. Is the answer smaller than the given numbers (groups, pieces) or larger (time together, total bought)? This alone eliminates choices.
  • Plug in answer choices. For "which number is prime?", divide each choice by 3 and 7. For a missing-number problem, test each choice against all the conditions.
  • Multiply back. Any factorization or GCF/LCM answer can be checked in seconds by multiplying or dividing.
  • Read the last line twice. Multi-step problems ask for the number of tiles, trays or times — not the GCF or LCM itself.
  • Use units. Convert minutes to seconds (or feet to inches) before finding an LCM or GCF.
  • Time: these are quick if you know the rules. Spend about a minute; if stuck, list multiples or factors on your erasable note board.

Check yourself

Multiples and factorsNo calculator

Which of these numbers is prime?

Answer choices
Hint

A prime number has exactly two factors: 1 and itself. Try dividing each number by small primes: 3, 7, 11, 13.

Check yourself

Multiples and factorsNo calculatorFill in

What is the greatest common factor (GCF) of 24 and 36?

Type your answer in the box. Fractions like 3/4 are fine.

Hint

List the factors of each number, then find the largest one they share.

Summary

  • Factors divide into a number; multiples are what you get by counting by the number.
  • A prime has exactly two factors; 1 is not prime, and 2 is the only even prime.
  • Use divisibility rules and factor trees to find a prime factorization like 360 = 2^3 \times 3^2 \times 5.
  • GCF: shared primes, smaller powers — for splitting into identical groups. LCM: all primes, larger powers — for "together again" and "none left over".
  • For two numbers, \text{GCF} \times \text{LCM} = a \times b — useful for finding a missing number.
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