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Factors &
Multiples

Master the building blocks of number relationships: learn to identify factors, list multiples, and use them to solve problems on the SSAT.

SSAT Middle Level

What Is a Factor?

A factor of a whole number is any whole number that divides into it evenly, with no remainder. When you multiply two factors together, they produce the original number.

What Is a Multiple?

A multiple of a whole number is the result of multiplying that number by any positive whole number. Multiples keep going forever: there are infinitely many.

Seeing Factor Pairs Visually

Every factor has a partner. Together they form a factor pair that multiplies to give the original number. We can arrange objects into rectangular arrays to see them.

Factor Pairs of 12
\(1 \times 12\)
\(2 \times 6\)
\(3 \times 4\)
\(4 \times 3\)
 Factor pairs: (1, 12) (2, 6) (3, 4) All factors of 12: {1, 2, 3, 4, 6, 12}

The Divisibility Rule for Factors

\(b = a \times k\)   (remainder = 0)
\(a\) is a factor of \(b\)
\(b\) is a multiple of \(a\)

Finding All Factors Systematically

Finding factors of 36 1 36
\(1 \times 36\)
 2 18
\(2 \times 18\)
 3 12
\(3 \times 12\)
 4 9
\(4 \times 9\)
 6 6
\(6 \times 6\)
 STOP: 6 × 6 means pairs start repeating!

Factors vs. Multiples

Factors
Divide into the number evenly. Always smaller than or equal to the number. The list is finite.
Multiples
Result from multiplying the number. Always larger than or equal to the number. The list is infinite.
For the number 6: Factors: 1 2 3 6 (only 4!) Multiples: 6 12 18 24 30 ... (infinite!)

GCF and LCM

Two powerful tools built from factors and multiples appear frequently on the SSAT:

GCF
Greatest Common Factor: the largest factor shared by two or more numbers.
LCM
Least Common Multiple: the smallest multiple shared by two or more numbers.

How to Find Factors and Multiples

1
To list factors: Start dividing the number by 1, 2, 3, ... and record every divisor that gives remainder 0. Stop when pairs begin repeating.
2
To list multiples: Multiply the number by 1, 2, 3, 4, 5, ... The list continues infinitely, so list as many as needed.
3
To find GCF: List all factors of both numbers, then pick the greatest one they share.
4
To find LCM: List multiples of both numbers until you find the smallest one they share.

Example 1: Listing Factors of 24

Find all factors of 24.

Systematic Division Check
\(24 \div 1 = 24\)
 ✓ pair: (1, 24)
\(24 \div 2 = 12\)
 ✓ pair: (2, 12)
\(24 \div 3 = 8\)
 ✓ pair: (3, 8)
\(24 \div 4 = 6\)
 ✓ pair: (4, 6)
\(24 \div 5 = 4.8\)
 ✕ not a whole number STOP at 5 next would be 6, already found Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24} 8 factors total

Example 2: Finding the GCF

Find the GCF of 18 and 24.

Factors of 18 1, 2, 3, 6, 9, 18 Factors of 24 1, 2, 3, 4, 6, 8, 12, 24 Common factors 1 2 3 6 GCF = 6

Example 3: Finding the LCM

Find the LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ... Multiples of 6: 6, 12, 18, 24, 30, 36, ... Common multiples: 12, 24, 36, ... LCM = 12

The Prime Factorization Shortcut

Breaking numbers into prime factors gives a fast way to find GCF and LCM without listing everything.

18 18 2 9 3 3
\(18 = 2 \times 3 \times 3\)
 24 24 2 12 2 6 2 3
\(24 = 2 \times 2 \times 2 \times 3\)

GCF and LCM from Primes

GCF Rule
Take the lowest power of each shared prime factor.
LCM Rule
Take the highest power of each prime factor from either number.

Common Mistakes to Avoid

✕ Swapping Factors and Multiples
Saying "12 is a factor of 3" instead of "3 is a factor of 12." Remember: factors go into the number, multiples come out.
✕ Forgetting 1 and the Number Itself
Every number has 1 and itself as factors. When listing factors, always start with 1 and end with the number.
✕ Confusing GCF and LCM
GCF is the greatest shared factor (always \( \le \) both numbers). LCM is the least shared multiple (always \( \ge \) both numbers).

Summary Comparison

Factors Multiples
Definition Divide evenly into the number Result of multiplying the number
Direction \(\le\) the number \(\ge\) the number
Count Finite (limited list) Infinite (never-ending)
Smallest Always 1 The number itself
Related tool GCF (Greatest Common Factor) LCM (Least Common Multiple)
How to find Divide systematically from 1 Multiply by 1, 2, 3, ...

Question 1 of 5

Which of the following is NOT a factor of 48?

Question 2 of 5

What is the least common multiple (LCM) of \(8\) and \(12\)?

Question 3 of 5

How many factors does the number 30 have?

Question 4 of 5

What is the greatest common factor (GCF) of \(36\) and \(48\)?

Question 5 of 5

Which of the following is a common multiple of both \(5\) and \(7\)?