1. Key Definitions

• Factor (or divisor): A number that divides another number exactly.
  Example: Factors of 12 are 1, 2, 3, 4, 6, 12…

• Multiple: The result of multiplying a number by an integer.
  Example: Multiples of 6 are 6, 12, 18, 24,…

2. Prime Factorization

• Breaking down a number into a product of primes.
  Example:
  $$84=2^2\cdot 3\cdot 7$$

• Prime factorization is the foundation for:

  • GCF (Greatest Common Factor)

  • LCM (Least Common Multiple)

  • Counting factors

3. Greatest Common Factor (GCF)

• GCF is the largest number that divides two numbers.

• To find the GCF of two numbers:

  • Factor both numbers into primes.

  • Take the lowest power of each common prime. 

Example: Find the GCF of 36 and 60

$$36= 2^2 \cdot 3^2$$ and $$60 = 2^2 \cdot 3 \cdot 5$$
$$GCF=2^2 \cdot 3 = 12$$

4. Least Common Multiple (LCM)

• The smallest number that is a multiple of two numbers.

• To find the GCF of two or multiple numbers:

  1- Factor both into primes. 

         2- Take the highest power of each prime.

Example:

$$12=2^2 \cdot 3$$ and $$18 = 2 \cdot 3^2$$
$$LCM=2^2 \cdot 3^2 = 36$$

5. Divisibility Rules (Must memorize for the ACT)

• 2: Last digit even

• 3: Sum of digits divisible by 3

• 4: Last 2 digits divisible by 4

• 5: Last digit 0 or 5

• 6: Divisible by both 2 and 3

• 8: Last 3 digits divisible by 8

• 9: Sum of digits divisible by 9

• 10: Last digit 0

6. Counting Factors

• For prime factorization n= p_1^{a} \cdot p_2^{b} \cdot \dots

• Total number of factors = $\dots$

Example:

$$84= 2^2 \cdot 3^1 \cdot 7^1$$
Factors = $12$

7. Common ACT Question Types

1. Finding GCF/LCM of two numbers.

2. Counting divisors of a given number.

3. Divisibility test questions.

4. Word problems:

   • “A number is divisible by both 6 and 8. What is the smallest such number?” (LCM problem).

   • “How many positive integers less than 100 are multiples of 6 but not multiples of 9?” (Inclusion-exclusion).

   • “If n is divisible by 12 and 15, what’s the least value n can be?” (LCM again).

8. Common ACT Pitfalls

• Forgetting 1 is a factor of every number.

• Confusing factors (smaller) with multiples (larger).

• Stopping prime factorization too early.

• Overlooking the wording:

  • “What is the least positive integer divisible by both…” = LCM.

  • “What is the greatest integer that divides both…” = GCD.

9. Summary Formula Sheet for Factors & Multiples:

• GCD: lowest powers of primes.

• LCM: highest powers of primes.

• Number of factors: (exponents + 1) multiplied.

• Relationship: GCD × LCM = Product of numbers.