Factors Multiples and Primes — ACT Math Guide

Q: What You Need to Know

• **Factors** are numbers that divide evenly into another number (12 has factors: 1, 2, 3, 4, 6, 12) • **Multiples** are products of a number and integers (multiples of 3: 3, 6, 9, 12, 15...) • **Prime numbers** have exactly two factors: 1 and themselves (2, 3, 5, 7, 11, 13...) • **Composite numbers

Factors multiples and primes ACT questions test your understanding of how numbers relate to each other. These problems ask you to find factors, identify multiples, or determine if numbers are prime or composite. The ACT math section includes 2-4 questions on this topic out of the 60 questions you'll face in 60 minutes. You've got the foundation for this from middle school — now let's make sure you can handle these efficiently under test pressure.

What You Need to Know

Factors are numbers that divide evenly into another number (12 has factors: 1, 2, 3, 4, 6, 12)

Multiples are products of a number and integers (multiples of 3: 3, 6, 9, 12, 15...)

Prime numbers have exactly two factors: 1 and themselves (2, 3, 5, 7, 11, 13...)

Composite numbers have more than two factors (4, 6, 8, 9, 10, 12...)

Greatest Common Factor (GCF) is the largest number that divides into two or more numbers

Least Common Multiple (LCM) is the smallest positive number that's a multiple of two or more numbers

📐 KEY FORMULA: For two numbers a and b: GCF(a,b) × LCM(a,b) = a × b

⏱️ ACT TIME TIP: Memorize primes up to 30 (2,3,5,7,11,13,17,19,23,29) — saves precious seconds on factor questions

How to Solve Factors Multiples and Primes on the ACT

Example Question 1 — Easy/Medium Difficulty

What is the greatest common factor of 48 and 72?

A) 6

B) 12

C) 18

D) 24

E) 36

Solution:

Step 1: Find factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Step 2: Find factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Step 3: Identify the greatest common factor: 24

✅Answer: D — 24 is the largest number that divides evenly into both 48 and 72

Example Question 2 — Hard Difficulty

If n is a positive integer and 2n + 1 is prime, which of the following could be the value of n?

A) 15

B) 20

C) 25

D) 30

E) 35

Solution:

Step 1: Test each option by substituting n and checking if 2n + 1 is prime

Step 2: For n = 15: 2(15) + 1 = 31 (prime ✓)

Step 3: For n = 20: 2(20) + 1 = 41 (prime ✓)

Step 4: Check other options: 51 (divisible by 3), 61 (prime), 71 (prime)

✅Answer: A — When n = 15, we get 2(15) + 1 = 31, which is prime

Common ACT Math Mistakes to Avoid

❌Mistake: Confusing factors and multiples (saying 24 is a multiple of 48)

✅Fix: Remember factors divide INTO a number, multiples come FROM multiplying

❌Mistake: Forgetting that 1 is not prime (it only has one factor: itself)

✅Fix: Prime numbers need exactly TWO factors — 1 and the number itself

❌Mistake: Missing negative factors when the question allows them

✅Fix: Read carefully — some ACT questions specify "positive factors" while others don't

❌Mistake: Calculating LCM by just multiplying the two numbers together

✅Fix: Use the formula LCM(a,b) = (a × b) ÷ GCF(a,b) for efficiency

Practice Question — Try It Yourself

Which of the following numbers has exactly 6 positive factors?

A) 16

B) 18

C) 20

D) 24

E) 32

Show Answer

Answer: C — 20 has factors 1, 2, 4, 5, 10, 20 (exactly 6 factors)

Key Takeaways for the ACT

Prime numbers less than 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 — memorize these

Factor pairs work both ways: if a is a factor of b, then b is a multiple of a

Use prime factorization for complex GCF and LCM problems — it's faster than listing

The ACT math section allows calculators, but mental math is often quicker for basic factor work

With 60 questions in 60 minutes, spend no more than 90 seconds on straightforward factor problems