Factors & Multiples
A complete, test-focused breakdown of every Factors & Multiples concept on the ACT — from prime factorization and divisibility rules to GCF, LCM, and number theory applications — with worked examples, visual tools, and strategy notes for test day.
In This Lesson
Factors & Multiples — Definitions
These two concepts are mirrors of each other. Confusing them is one of the most common ACT errors — nail the distinction now.
Factors of 12: 1, 2, 3, 4, 6, 12
Because 12÷1=12, 12÷2=6, 12÷3=4, 12÷4=3, 12÷6=2, 12÷12=1 — all whole numbers, zero remainders.
Multiples of 4: 4, 8, 12, 16, 20, …
Because 4×1=4, 4×2=8, 4×3=12, 4×4=16 — infinitely many multiples.
"Factors are Few, Multiples are Many."
Any integer has a finite number of factors but infinite multiples.
12 has exactly 6 factors. 12 has infinitely many multiples.
Factor Pairs
Factors always come in pairs that multiply to give the original number. List pairs systematically by testing divisors from 1 upward until you reach the square root — beyond that, pairs repeat.
Square root shortcut: You only need to test divisors up to √n. Once you pass the square root, you're just finding the same pairs in reverse. This saves time on the ACT when n is large.
Key Properties to Know
- Every integer is a factor of itself and has 1 as a factor.
- Every positive integer is a multiple of 1 and a multiple of itself.
- If a is a factor of b, then b is a multiple of a — and vice versa.
- If a is a factor of b AND b is a factor of c, then a is a factor of c (transitivity).
- If a and b are both factors of n, then (a+b) may or may not be a factor of n.
Divisibility Rules
Divisibility rules let you instantly check whether a number is divisible by another — without doing long division. On the ACT, these save significant time, especially on factor, GCF, and prime factorization problems.
| Divisible by | Rule | Example — 2,184 |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 (even number) | ✓ Last digit 4 → divisible |
| 3 | Sum of all digits is divisible by 3 | ✓ 2+1+8+4 = 15, 15÷3=5 → divisible |
| 4 | Last two digits form a number divisible by 4 | ✓ Last two: 84, 84÷4=21 → divisible |
| 5 | Last digit is 0 or 5 | ✗ Last digit 4 → not divisible |
| 6 | Divisible by BOTH 2 AND 3 | ✓ Even + digit sum 15 → divisible |
| 7 | No easy rule — divide directly or use prime factorization | 2184 ÷ 7 = 312 → divisible |
| 8 | Last three digits form a number divisible by 8 | ✓ Last three: 184, 184÷8=23 → divisible |
| 9 | Sum of all digits is divisible by 9 | ✗ Digit sum = 15, 15÷9 not whole → not divisible |
| 10 | Last digit is 0 | ✗ Last digit 4 → not divisible |
| 11 | Alternating sum of digits (left to right) is divisible by 11 or equals 0 | ✓ 2−1+8−4 = 5 → not divisible (2,184÷11 = 198.5) |
| 12 | Divisible by BOTH 3 AND 4 | ✓ Div by 3 and div by 4 → divisible by 12 |
Combine rules: Divisibility by 6 = divisible by 2 AND 3. Divisibility by 12 = divisible by 3 AND 4. Divisibility by 15 = divisible by 3 AND 5. These compound rules often appear on the ACT without being explicitly stated.
Prime & Composite Numbers
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16…
1 is NEITHER prime nor composite — it has only one factor (itself).
2 is the ONLY even prime number — all other even numbers are composite.
All primes greater than 3 are of the form 6k±1 (but not all 6k±1 are prime).
Primes Under 100 — The Sieve of Eratosthenes
There are 25 prime numbers under 100. Memorize those under 50 — they appear constantly on the ACT.
Blue = prime · Gray = composite · Light = 1 (neither)
Testing Whether a Number is Prime
To check if n is prime, test all prime divisors up to √n. If none divide evenly, n is prime.
Common mistakes: Students often call 1 prime (it isn't) or assume 51 is prime (51 = 3×17) or 57 is prime (57 = 3×19). Always check divisibility by 3 using the digit-sum rule first — it catches many "prime-looking" composites.
Prime Factorization
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be written as a unique product of prime numbers (up to order). This representation is called the prime factorization.
Prime factorization is the engine behind GCF, LCM, factor counting, and many ACT number theory problems.
Factor Tree Method
Split the number into any two factors, then keep factoring each branch until all leaves are prime.
Division Ladder Method
Divide repeatedly by the smallest prime that goes in evenly, working downward.
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
// Result: 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5
Always write the prime factorization with prime bases in ascending order with exponents:
n = p₁^a₁ × p₂^a₂ × p₃^a₃ × …
where p₁ < p₂ < p₃ < … are distinct primes and a₁, a₂, a₃ … are positive integers.
Use divisibility rules while factoring: Always test 2 first (is it even?), then 3 (digit sum), then 5 (ends in 0 or 5), then 7, then 11. This order minimizes the work on the ACT.
Greatest Common Factor (GCF)
The GCF (also called GCD — Greatest Common Divisor) of two or more numbers is the largest integer that divides all of them evenly.
Method 1 — List All Factors
Write out all factors of each number and find the largest one they share. Works well for small numbers.
Method 2 — Prime Factorization (Best for Large Numbers)
Factor each number. Take the lowest power of every prime factor they share in common.
Method 3 — Euclidean Algorithm (Fastest for Two Large Numbers)
Repeatedly replace the larger number with the remainder of dividing the larger by the smaller. Stop when the remainder is 0 — the last nonzero remainder is the GCF.
When to use which method: Small numbers → list factors. Medium numbers → prime factorization. Large "ugly" numbers → Euclidean algorithm. On the ACT, prime factorization handles most cases since numbers are rarely huge.
GCF of Three or More Numbers
Use prime factorization: take the lowest exponent of each prime that appears in ALL numbers.
120 = 2³ × 3 × 5
180 = 2² × 3² × 5
240 = 2⁴ × 3 × 5
GCF = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60
Least Common Multiple (LCM)
The LCM of two or more numbers is the smallest positive integer that is a multiple of all of them. Essential for adding fractions, scheduling problems, and cycle questions on the ACT.
Method 1 — List Multiples
Write out multiples of the larger number until you find one divisible by the smaller. Quick for small numbers.
Method 2 — Prime Factorization (Best Method)
Factor each number. Take the highest power of every prime factor that appears in any of the numbers.
GCF vs. LCM confusion trap: GCF uses the lowest exponent of shared primes. LCM uses the highest exponent of all primes. Students frequently swap these. A memory trick: GCF → Go low. LCM → Lift high.
The GCF × LCM Relationship
For any two positive integers a and b:
GCF(a, b) × LCM(a, b) = a × b
This means: if you know any three of the four values, you can find the fourth instantly.
Finding Two Numbers from GCF and LCM
If you know the GCF and LCM of two numbers, you can find candidate pairs. Write each number as GCF × (some factor), where the two "some factors" must be coprime (GCF of 1) and their product = LCM ÷ GCF.
Counting the Number of Factors
The ACT sometimes asks how many factors a number has without asking you to list them. There is a direct formula using prime factorization.
If n = p₁^a₁ × p₂^a₂ × p₃^a₃ × …
Then the total number of factors of n =
(a₁ + 1)(a₂ + 1)(a₃ + 1) …
Add 1 to each exponent, then multiply those results together.
ACT application: "Which of the following has exactly 6 factors?" → You need n with (a+1)(b+1)… = 6. Options: exponent pattern 5 → gives 6 (one prime, e.g. p⁵), or exponent pattern 2,1 → gives (2+1)(1+1)=6 (two primes, e.g. p²×q). Check both patterns quickly.
Finding Numbers with a Specific Factor Count
| Target Factor Count | Exponent Pattern | Smallest Example |
|---|---|---|
| 2 factors | n = p¹ | 2 (prime) |
| 3 factors | n = p² | 4 = 2² |
| 4 factors | n = p³ or p¹×q¹ | 8 or 6 |
| 6 factors | n = p⁵ or p²×q¹ | 32 or 12 |
| 8 factors | n = p⁷ or p³×q¹ or p¹×q¹×r¹ | 128 or 24 or 30 |
| Odd factor count | All exponents must be even | n must be a perfect square |
ACT Applications & Word Problems
Tiling / Cutting / Splitting Problems → GCF
Whenever a problem asks for the largest equal pieces, the fewest cuts, or the biggest tile that fits without gaps, the answer is a GCF.
Scheduling / Cycle Problems → LCM
Whenever a problem asks when will two events coincide again, or what is the earliest time both occur together, the answer is an LCM.
Distributing / Grouping Problems → GCF
When you want to split multiple quantities into the largest equal groups with nothing left over, use GCF.
Smallest Number Satisfying Multiple Divisibility Conditions → LCM
"Remainder" Problems — A Hidden Divisibility Pattern
Problems of the type "when divided by 5 leaves remainder 2, when divided by 7 leaves remainder 2" — the common remainder clue means the answer is LCM + remainder.
Test Day Strategy
Top ACT Traps to Avoid
- 1 is not prime. It has only one factor — itself. Every ACT that asks about primes tries to catch you here.
- 2 is prime — the only even prime. Don't exclude it.
- Swapping GCF and LCM: GCF is always ≤ both numbers; LCM is always ≥ both.
- For GCF, use the lowest exponents of shared primes. For LCM, use the highest exponents of all primes.
- Forgetting to check if a "prime-looking" number like 51 or 91 is actually composite (51=3×17, 91=7×13).
- Answering "how many factors" by listing instead of using the formula — wastes time on large numbers.
Decision Framework
- "Largest equal piece / group" → GCF
- "Next time both/all coincide" → LCM
- "Smallest divisible by all" → LCM
- "Fewest cuts with no waste" → GCF
- "How many factors does n have?" → factor-count formula
- "Is n prime?" → test primes up to √n
Must-Know Quick Reference
| Concept | Formula / Rule | Key Detail |
|---|---|---|
| GCF (prime factorization) | Shared primes, LOWEST exponents | GCF ≤ min(a, b) |
| LCM (prime factorization) | All primes, HIGHEST exponents | LCM ≥ max(a, b) |
| GCF × LCM relationship | GCF(a,b) × LCM(a,b) = a × b | Works for 2 numbers only |
| Number of factors | (a₁+1)(a₂+1)… from n=p₁^a₁×p₂^a₂… | Odd count → perfect square |
| Divisibility by 3 | Sum of digits divisible by 3 | Works for 9 too (sum div by 9) |
| Divisibility by 6 | Divisible by 2 AND 3 | Compound rule |
| Primality test | Test prime divisors up to √n | If none found → prime |
| Primes under 20 | 2, 3, 5, 7, 11, 13, 17, 19 | 8 primes — memorize all |
Put It to the Test
You've covered every concept. Lock it in with ACT-style timed questions. Start with Quiz 1 and progress through Quiz 3 — each set increases in difficulty and problem variety.