Factors and Multiples — Determine factors and multiples of a number. - SHSAT Math

SECTION 1

Historical Context & Motivation

People have been breaking numbers apart and building them up for thousands of years. Ancient civilizations needed to divide land, share harvests, and organize armies. To do this, they had to understand which numbers divide evenly into other numbers. This is exactly what factors and multiples are all about.

~2000 BCE • Babylonian Number Tables

Ancient Babylonians carved clay tablets listing multiplication facts. These early "times tables" helped traders and builders work with factors and multiples every day.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid wrote a famous book that defined prime numbers and showed how to find the greatest common factor of two numbers. His method is still used today!

~240 BCE • Sieve of Eratosthenes

Eratosthenes invented a simple method to find all prime numbers up to any limit. His "sieve" works by crossing out multiples of each prime, one by one.

1801 • Gauss and Number Theory

Carl Friedrich Gauss published a groundbreaking book about integers. He proved that every whole number greater than 1 can be broken into a unique set of prime factors.

So why should you care about factors and multiples on the SHSAT? Many test questions ask you to find common factors, least common multiples, or decide if one number divides evenly into another. Understanding these ideas is like having a secret key that unlocks a whole family of problems.

SECTION 2

Core Principles & Definitions

Before you can solve problems, you need to know the key vocabulary. Let's break it down into clear ideas.

Factor

A factor of a number is a whole number that divides into it evenly (with no remainder). For example, 4 is a factor of 12 because 12 ÷ 4 = 3 exactly.

Multiple

A multiple of a number is the result you get when you multiply that number by any whole number. For example, 12 is a multiple of 4 because 4 × 3 = 12.

Factor Pairs

Factors always come in pairs. If 3 × 4 = 12, then both 3 and 4 are factors of 12. Finding one factor automatically gives you its partner.

Divisibility

We say "a is divisible by b" when a ÷ b leaves no remainder. Divisibility is just another way to talk about factors.

Factors Are Finite, Multiples Are Infinite

Every number has a limited (finite) list of factors, but an unlimited (infinite) list of multiples. For instance, 6 has only four factors (1, 2, 3, 6) but multiples of 6 go on forever (6, 12, 18, 24, …).

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KEY TAKEAWAY

Think of factors and multiples like a family tree. Factors are the "parents" that multiply together to create the number, while multiples are the "children" that the number produces when it multiplies with others. If 3 × 4 = 12, then 3 and 4 are factors (parents) of 12, and 12 is a multiple (child) of both 3 and 4.

SECTION 3

Visual Explanation — Factor Pairs of 24

The best way to find all the factors of a number is to look for pairs. Let's use the number 24 as our example. We start from 1 and work upward, checking which numbers divide evenly into 24.

Each colored arc connects a factor pair — two numbers that multiply together to make 24. Notice that we stop when the pairs start to repeat. Since 4 × 6 = 24 and 5 does not divide evenly into 24, we know we have found all four pairs. That gives us eight total factors.

Here's the trick: you only need to check numbers up to the square root of your target number. The square root of 24 is about 4.9, so once you reach 5 and see it doesn't divide evenly, you're done. Every factor pair has one number below the square root and one above it.

SECTION 4

Mathematical Framework

Let's put the ideas of factors and multiples into simple formulas so you can use them on any problem.

FACTOR DEFINITION

If a ÷ b = whole number (no remainder), then b is a factor of a.

Example: 18 ÷ 3 = 6 (no remainder), so 3 is a factor of 18.

MULTIPLE DEFINITION

Multiple of n = n × k, where k = 1, 2, 3, 4, …

Example: Multiples of 7 are 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28, and so on forever.

GREATEST COMMON FACTOR (GCF)

GCF(a, b) = the largest factor shared by both a and b

The GCF (also called the Greatest Common Divisor) is the biggest number that divides evenly into both numbers. For 12 and 18, the shared factors are 1, 2, 3, and 6, so GCF(12, 18) = 6.

LEAST COMMON MULTIPLE (LCM)

LCM(a, b) = the smallest multiple shared by both a and b

The LCM is the smallest number that both a and b divide into evenly. For 4 and 6: multiples of 4 are 4, 8, 12, 16, … and multiples of 6 are 6, 12, 18, … The first overlap is 12, so LCM(4, 6) = 12.

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SHSAT Tip

A handy shortcut connects GCF and LCM: GCF(a, b) × LCM(a, b) = a × b. If you know one, you can quickly find the other. For example, GCF(4, 6) = 2 and 4 × 6 = 24, so LCM(4, 6) = 24 ÷ 2 = 12.

SECTION 5

Divisibility Rules — Your Speed Shortcuts

On the SHSAT, you don't always have time for long division. Divisibility rules let you quickly check if a number is a factor of another number just by looking at its digits. Memorize these rules — they save valuable seconds.

This chart shows the most important divisibility rules for digits 2 through 10. The example at the bottom shows how to combine rules: to check divisibility by 6, you test for both 2 and 3.

Let's test the number 360 as a quick example. It ends in 0, so it's divisible by 2, 5, and 10. The digit sum is 3 + 6 + 0 = 9, which is divisible by both 3 and 9. Since it passes the rules for 2 and 3, it is also divisible by 6. The last two digits "60" are divisible by 4 (60 ÷ 4 = 15). The last three digits "360" are divisible by 8 (360 ÷ 8 = 45). So 360 is divisible by every number from 2 to 10 except 7!

SECTION 6

Worked Example — Find All Factors and Some Multiples of 36

Let's walk through a full problem, step by step. This is the kind of question you might see on the SHSAT.

Find all factors of 36 and list the first six multiples of 36.

1
Step 1 — Start with 1

Every number has 1 and itself as factors. So we start our factor list with 1 and 36. We write the pair: 1 × 36 = 36.

Pair: (1, 36)
2
Step 2 — Check 2

Is 36 divisible by 2? The last digit is 6 (even), so yes. 36 ÷ 2 = 18. We get the pair 2 × 18 = 36.

Pair: (2, 18)
3
Step 3 — Check 3

Is 36 divisible by 3? The digit sum is 3 + 6 = 9, which is divisible by 3, so yes. 36 ÷ 3 = 12. Pair: 3 × 12 = 36.

Pair: (3, 12)
4
Step 4 — Check 4

36 ÷ 4 = 9, with no remainder. Pair: 4 × 9 = 36.

Pair: (4, 9)
5
Step 5 — Check 5

36 does not end in 0 or 5, so 5 is NOT a factor. Skip it.
6
Step 6 — Check 6

36 ÷ 6 = 6. This is a special case: the pair is 6 × 6 = 36. Since both numbers in the pair are the same, 6 is the square root of 36. We stop here because the next number (7) is larger than 6.

Pair: (6, 6) — we've found them all!
7
Step 7 — List all factors

Collect all the numbers from every pair and sort them in order.

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
8
Step 8 — List the first six multiples

Multiply 36 by 1, 2, 3, 4, 5, and 6: 36 × 1 = 36, 36 × 2 = 72, 36 × 3 = 108, 36 × 4 = 144, 36 × 5 = 180, 36 × 6 = 216.

First six multiples of 36: 36, 72, 108, 144, 180, 216

SECTION 7

Comparing Methods for Finding Factors

There are several ways to find factors and multiples. Each method has strengths and weaknesses. On the SHSAT, you should pick the method that is fastest for the number you're given.

Method	How It Works	Best For	Drawback
Factor Pairs	Divide by 1, 2, 3, … up to the square root. Record each pair.	Numbers under 100; gives you every factor.	Can be slow for large numbers.
Divisibility Rules	Use digit patterns to check 2, 3, 4, 5, 6, 8, 9, 10 quickly.	Large numbers; quick yes/no checks.	Doesn't cover every possible factor (like 7 or 11).
Prime Factorization	Break the number into prime factors using a factor tree.	Finding GCF and LCM of two or more numbers.	Takes more steps; you must know your primes.
Listing Multiples	Multiply the number by 1, 2, 3, … until you find what you need.	Finding the LCM of small numbers.	Lists can get long for larger numbers.

Comparison of four common methods for working with factors and multiples

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KEY TAKEAWAY

Think of finding factors like unpacking a suitcase. The factor pair method opens every pocket one by one. Divisibility rules are like an X-ray scanner — they let you peek inside without opening anything. Prime factorization takes the suitcase apart completely so you can see every tiny piece. Pick the right tool for the job!

SECTION 8

Connection to GCF and LCM

Once you're comfortable finding factors and multiples of a single number, the next level is comparing two (or more) numbers. That's where the Greatest Common Factor (GCF) and Least Common Multiple (LCM) come in. The SHSAT loves to test these.

	GCF (Greatest Common Factor)	LCM (Least Common Multiple)
Question it answers	What is the LARGEST number that divides evenly into both?	What is the SMALLEST number that both divide into evenly?
Direction	Look DOWN — find shared factors.	Look UP — find shared multiples.
Example (12 and 18)	Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. GCF = 6.	Multiples of 12: 12, 24, 36, … Multiples of 18: 18, 36, 54, … LCM = 36.
Real-life use	Splitting items into equal groups (e.g., dividing 12 red and 18 blue marbles into identical bags).	Finding when events sync up (e.g., two buses that leave every 12 and 18 minutes).
Shortcut formula	Use prime factorization: take the LOWEST power of each shared prime.	Use prime factorization: take the HIGHEST power of every prime that appears.

Side-by-side comparison of GCF and LCM

As you move into more advanced math, factors and multiples lead to simplifying fractions (using the GCF), adding fractions with unlike denominators (using the LCM), and even factoring algebraic expressions in high school. Every one of those skills starts right here.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

True or false: 7 is a factor of 49 AND 49 is a multiple of 7. Explain why both statements are connected.

Show answer

True — both statements are true and they express the same relationship from different directions. Because 7 × 7 = 49, we know 7 divides evenly into 49, making 7 a factor of 49. At the same time, 49 is the result of multiplying 7 by a whole number (7), making 49 a multiple of 7. Whenever a is a factor of b, b is automatically a multiple of a. This two-way connection is the heart of the factor–multiple relationship.

Question 2: BASIC CALCULATION

List all the factors of 42.

Show answer

Start from 1 and work up to the square root of 42 (about 6.5). Check each: 1 × 42 = 42 ✓, 2 × 21 = 42 ✓, 3 × 14 = 42 ✓, 42 ÷ 4 = 10.5 ✗, 42 ÷ 5 = 8.4 ✗, 6 × 7 = 42 ✓. Since 7 > 6.5, we stop. The complete list is: 1, 2, 3, 6, 7, 14, 21, 42. That's 8 factors in total.

Question 3: INTERMEDIATE

Find the GCF and LCM of 24 and 36.

Show answer

First, list the factors of each number. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The common factors are 1, 2, 3, 4, 6, and 12. The greatest is 12, so GCF(24, 36) = 12. For the LCM, use the shortcut: LCM = (24 × 36) ÷ GCF = 864 ÷ 12 = 72. You can verify: 72 ÷ 24 = 3 and 72 ÷ 36 = 2, both whole numbers. So LCM(24, 36) = 72.

Question 4: APPLIED

A teacher has 48 pencils and 60 erasers. She wants to make identical supply bags with no items left over. What is the greatest number of bags she can make, and how many pencils and erasers go in each bag?

Show answer

This is a GCF problem because we are splitting items into equal groups. Find GCF(48, 60). Factors of 48 include 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 60 include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The greatest common factor is 12. So the teacher can make 12 bags. Each bag gets 48 ÷ 12 = 4 pencils and 60 ÷ 12 = 5 erasers. No items are left over.

Question 5: CRITICAL THINKING

A number N is between 1 and 100. It has exactly 3 factors. What are all the possible values of N? Explain your reasoning.

Show answer

A number with exactly 3 factors must be the square of a prime number. Here's why: if N = p², then its factors are 1, p, and p². No other arrangement gives exactly three factors. (A prime itself has only 2 factors, and any non-square composite has at least 4.) The primes whose squares are under 100 are 2, 3, 5, and 7. So N can be 2² = 4, 3² = 9, 5² = 25, or 7² = 49. Check: factors of 4 are {1, 2, 4}, factors of 9 are {1, 3, 9}, factors of 25 are {1, 5, 25}, factors of 49 are {1, 7, 49}. Each has exactly 3 factors. The answer is N ∈ {4, 9, 25, 49}.

SUMMARY

Lesson Summary

Summary

A factor is a whole number that divides evenly into another number, while a multiple is the result of multiplying a number by any whole number. Every number has a finite list of factors but an infinite list of multiples. To find all factors, use the factor pair method — test divisors from 1 up to the square root and record each pair. Speed up your work with divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10.

When comparing two numbers, the Greatest Common Factor (GCF) is the largest factor they share, and the Least Common Multiple (LCM) is the smallest multiple they share. Remember the shortcut: GCF × LCM = product of the two numbers. These skills are essential for simplifying fractions, solving word problems, and tackling many SHSAT questions with confidence.