Factors & Multiples is a core ACT Math topic that shows up on its own and inside other problems, like simplifying fractions or finding a common denominator. This free, complete lesson covers prime numbers and prime factorization, divisibility rules, the greatest common factor (GCF), and the least common multiple (LCM). Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. Every composite number can be broken down into a unique product of prime numbers, its prime factorization.
Worked Example: Find the prime factorization of 84.
84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7
So 84 = 2² × 3 × 7.
1. Which of the following is a prime number?
Explanation: 29 has no factors other than 1 and itself. 21 = 3 × 7, 33 = 3 × 11, and 51 = 3 × 17, so all three are composite.
2. What is the prime factorization of 60?
Explanation: 60 = 4 × 15 = 2² × 3 × 5.
3. What is the prime factorization of 45?
Explanation: 45 = 9 × 5 = 3² × 5. Choices B and C aren't fully broken down into primes, and choice D equals 15, not 45.
4. How many distinct prime factors does 90 have?
Explanation: 90 = 2 × 3² × 5. The distinct prime factors are 2, 3, and 5, three of them.
5. Which of the following numbers is composite?
Explanation: 27 = 3³, which has factors other than 1 and itself, so it's composite. 17, 19, and 23 are all prime.
6. If m is even and n is odd, is m² + n² even or odd?
Explanation: An even number squared is even (m² is even), and an odd number squared is odd (n² is odd). Even + odd is always odd.
Divisibility rules let you check whether one number divides evenly into another without doing long division.
| Divisible by | Rule |
|---|---|
| 2 | The number is even (ends in 0, 2, 4, 6, or 8) |
| 3 | The sum of the digits is divisible by 3 |
| 4 | The last two digits form a number divisible by 4 |
| 5 | The number ends in 0 or 5 |
| 6 | The number is divisible by both 2 and 3 |
| 9 | The sum of the digits is divisible by 9 |
| 10 | The number ends in 0 |
Worked Example: Is 4,368 divisible by 3?
Add the digits: 4 + 3 + 6 + 8 = 21. Since 21 is divisible by 3, so is 4,368.
1. Is 5,742 divisible by 3?
Explanation: Digit sum: 5 + 7 + 4 + 2 = 18, which is divisible by 3, so 5,742 is too.
2. Which of the following numbers is divisible by 4?
Explanation: Check the last two digits of each: 16 (from 316) is divisible by 4. 22, 30, and 15 are not.
3. Is 4,995 divisible by 9?
Explanation: Digit sum: 4 + 9 + 9 + 5 = 27, which is divisible by 9, so 4,995 is too.
4. Which of the following numbers is divisible by 6?
Explanation: 348 is even, and its digit sum (3 + 4 + 8 = 15) is divisible by 3, so it's divisible by both 2 and 3, and therefore by 6. The others fail at least one of those two tests.
The greatest common factor of two or more numbers is the largest number that divides evenly into all of them. Find it by comparing the prime factorizations and multiplying the shared prime factors, using the lowest power each shares.
Worked Example: Find the GCF of 36 and 48.
36 = 2² × 3². 48 = 2⁴ × 3.
The shared factors are 2² and 3¹.
GCF = 2² × 3 = 12
1. What is the GCF of 24 and 40?
Explanation: 24 = 2³ × 3. 40 = 2³ × 5. The shared factor is 2³ = 8.
2. What is the GCF of 18 and 27?
Explanation: 18 = 2 × 3². 27 = 3³. The shared factor is 3² = 9.
3. What is the GCF of 15, 25, and 35?
Explanation: 15 = 3 × 5, 25 = 5², and 35 = 5 × 7. The only factor shared by all three is 5.
4. The GCF of two numbers is 6. If one number is 18, which of the following could be the other number?
Explanation: 18 = 2 × 3² and 24 = 2³ × 3, sharing 2 × 3 = 6 exactly. 36 shares more (GCF 18), 9 shares only 9, and 8 shares only 2.
5. Two numbers have a GCF of 1. What does this tell you about the numbers?
Explanation: A GCF of 1 means the numbers share no common prime factor. This describes "relatively prime" numbers, and neither number needs to be prime itself, for example, 8 and 9 are relatively prime.
The least common multiple of two or more numbers is the smallest number that all of them divide into evenly. Find it by taking each prime factor at its highest power appearing in any of the numbers.
Worked Example: Find the LCM of 8 and 12.
8 = 2³. 12 = 2² × 3.
Take the highest power of each prime: 2³ and 3¹.
LCM = 2³ × 3 = 24
1. What is the LCM of 6 and 9?
Explanation: 6 = 2 × 3. 9 = 3². LCM = 2 × 3² = 18.
2. What is the LCM of 4, 6, and 10?
Explanation: 4 = 2², 6 = 2 × 3, 10 = 2 × 5. LCM = 2² × 3 × 5 = 60.
3. Two bells ring every 12 minutes and every 18 minutes, respectively. If they ring together at 12:00 noon, when will they next ring together?
Explanation: The bells ring together again after LCM(12, 18) minutes. 12 = 2² × 3, 18 = 2 × 3², so LCM = 2² × 3² = 36 minutes, giving 12:36.
4. The LCM of two numbers is 24, and their GCF is 4. If one number is 8, what is the other number?
Explanation: For any two numbers, GCF × LCM equals the product of the numbers: 4 × 24 = 96. Since one number is 8, the other is 96 ÷ 8 = 12.
For small numbers, you can often list factors or multiples directly and compare. But prime factorization is the most reliable method for larger numbers or more than two numbers at once, and it's what makes the relationship GCF × LCM = product of the two numbers so useful.
Each rule comes from how our base-10 number system is structured. For example, since 100 is divisible by 4, only the last two digits of a number affect whether the whole number is divisible by 4, everything further left contributes a multiple of 100, which is already divisible by 4.
The most common use is finding a common denominator when adding or subtracting fractions, but LCM also appears directly in word problems about repeating events, like two things happening on a schedule that eventually line up again.
The Factors & Multiples quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.
