Apply ordering, factors, multiples, and exponents. - GED Math

SECTION 1

Historical Context & Motivation

Humans have needed to organize and compare numbers since the earliest civilizations. Ancient merchants had to arrange prices from cheapest to most expensive, builders needed to divide materials into equal groups, and astronomers worked with extraordinarily large numbers when describing distances between stars. The concepts of ordering, factors, multiples, and exponents evolved over thousands of years to solve exactly these kinds of practical problems.

~2000 BCE • Babylonian Number Tables

Ancient Babylonians created clay tablets listing multiplication tables and factor pairs, demonstrating an early understanding of divisibility.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid formalized rules for prime factors and the greatest common factor, ideas still used in today's math.

~200 BCE • Eratosthenes' Sieve

Eratosthenes developed an algorithm to find prime numbers by systematically eliminating multiples — a method still taught today.

1600s CE • Modern Exponent Notation

René Descartes introduced the superscript notation for exponents (like x²), replacing older, cumbersome ways of writing repeated multiplication.

Today, these concepts appear everywhere — from comparing prices at the grocery store to understanding how compound interest grows your savings. On the GED Mathematical Reasoning test, questions about ordering, factors, multiples, and exponents make up a significant portion of the Quantitative Problem Solving strand. Mastering them gives you a foundation for nearly every other math topic you will encounter.

SECTION 2

Core Principles & Definitions

Before we dive into calculations, let's establish the four key ideas you need. Each one builds on basic arithmetic you already know — addition, subtraction, multiplication, and division. Think of these concepts as tools in a toolbox: once you know which tool to reach for, the problems become much more manageable.

Ordering Numbers

Arranging numbers from least to greatest (ascending) or greatest to least (descending). Works with whole numbers, fractions, decimals, and negative numbers.

Factors

Whole numbers that divide evenly into a given number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Multiples

The results you get when you multiply a number by 1, 2, 3, and so on. The first five multiples of 4 are 4, 8, 12, 16, 20.

Exponents

A shorthand for repeated multiplication. In 5³, the base is 5 and the exponent is 3, meaning 5 × 5 × 5 = 125.

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KEY TAKEAWAY

Think of factors and multiples as opposite directions on the same road. Factors break a number down into smaller pieces (like cutting a pizza into equal slices), while multiples build a number up into larger amounts (like stacking identical boxes). If 3 is a factor of 12, then 12 is a multiple of 3 — same relationship, two perspectives.

SECTION 3

Visual Explanation — The Number Line & Factor Trees

A number line is one of the most powerful visual tools for understanding ordering. Numbers increase as you move to the right and decrease as you move to the left. This simple diagram makes it clear why −3 is less than 1, or why 0.25 comes before 0.75. The diagram below shows how different types of numbers — negatives, fractions, and positives — all fit on the same line.

The number line shows four highlighted values (−3, 0.5, 2, and 3) in their correct positions. The ordering from least to greatest reads left to right. Notice that negative numbers always fall to the left of zero.

When you encounter an ordering question on the GED, a quick mental number line can save you time — especially during the no-calculator Part 1. Place each value in your mind from left (smallest) to right (largest), and the correct order becomes clear.

SECTION 4

Mathematical Framework

Ordering Rules

To order numbers correctly, convert them to the same form. If you are comparing fractions and decimals, convert fractions to decimals first (divide the numerator by the denominator). For negative numbers, remember that a number farther from zero is smaller (for example, −5 < −2 because −5 is farther left on the number line).

Finding Factors

FACTOR TEST

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

Example: 24 ÷ 6 = 4, so 6 is a factor of 24. Always check pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6.

Finding Multiples

MULTIPLE FORMULA

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

Example: Multiples of 7 → 7, 14, 21, 28, 35, … The list never ends.

Exponent Rules

EXPONENT DEFINITION

bⁿ = b × b × b × … × b (n times)

b = base, n = exponent. Example: 2⁴ = 2 × 2 × 2 × 2 = 16.

KEY EXPONENT RULES

b⁰ = 1 | b¹ = b | bᵐ × bⁿ = bᵐ⁺ⁿ | (bᵐ)ⁿ = bᵐˣⁿ

Any nonzero number raised to the zero power equals 1. When multiplying the same base, add the exponents. When raising a power to a power, multiply the exponents.

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GED Tip — No-Calculator Section

Part 1 of the GED has no calculator. You should memorize common powers: 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 3² = 9, 3³ = 27, 4² = 16, 5² = 25, 5³ = 125, and 10² = 100. These come up frequently and save precious time.

SECTION 5

Detailed Breakdown — GCF, LCM, and Exponent Patterns

Two of the most tested ideas on the GED combine factors and multiples into specific tools: the Greatest Common Factor (GCF) and the Least Common Multiple (LCM). The GCF is the largest factor shared by two or more numbers — it helps when you need to simplify fractions or divide things into equal groups. The LCM is the smallest number that two or more numbers all divide into evenly — it's essential for adding fractions with different denominators or solving scheduling problems.

Factor trees break 24 and 36 down to their prime factors. The highlighted circles show the prime numbers at the "leaves" of each tree. The GCF uses the lowest shared powers (2² × 3¹ = 12), while the LCM uses the highest powers of all primes present (2³ × 3² = 72).

Power	Value	Meaning
2⁰	1	Any number to the 0 power is 1
2¹	2	The number itself
2²	4	2 × 2 — "two squared"
2³	8	2 × 2 × 2 — "two cubed"
2⁴	16	2 × 2 × 2 × 2
2⁵	32	2 × 2 × 2 × 2 × 2
10³	1,000	10 × 10 × 10 — adds three zeros

Common exponent values you should memorize for the GED

SECTION 6

Worked Example

Let's walk through a GED-style problem that combines several of these skills. This is the type of multi-step question you'll see on the actual test.

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Problem

A teacher is making supply bags for a class project. She has 48 pencils and 36 markers. She wants to divide them into identical bags so that each bag has the same number of pencils and the same number of markers, with nothing left over. What is the greatest number of bags she can make? How many pencils and markers will be in each bag?

Solution: Finding the GCF of 48 and 36

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Step 1 — Identify what we need

We need to split 48 pencils and 36 markers into equal groups with no leftovers. This means we need a number that divides evenly into both 48 and 36. We want the greatest such number, so we need the Greatest Common Factor (GCF).
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Step 2 — Find the prime factorization of each number

Break each number down using factor trees or repeated division. 48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3. Next, 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2² × 3².

48 = 2⁴ × 3¹ and 36 = 2² × 3²
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Step 3 — Take the lowest power of each shared prime

Both factorizations share the primes 2 and 3. For the base 2, the lowest power is 2² (from 36). For the base 3, the lowest power is 3¹ (from 48). So GCF = 2² × 3¹ = 4 × 3.

GCF = 12
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Step 4 — Answer the question

The teacher can make 12 bags. Each bag gets 48 ÷ 12 = 4 pencils and 36 ÷ 12 = 3 markers.

12 bags, each with 4 pencils and 3 markers

SECTION 7

Common Mistakes & How to Avoid Them

Many GED test-takers lose points not because they don't understand the concepts, but because they fall into predictable traps. The table below summarizes the most common mistakes and the corrections you should keep in mind.

Common Mistake	Why It's Wrong	Correct Approach
Confusing factors and multiples	Factors are smaller than or equal to the number; multiples are larger or equal.	Ask: does it divide IN (factor) or do I multiply OUT (multiple)?
Thinking 2³ = 6	Students multiply 2 × 3 instead of 2 × 2 × 2.	The exponent tells you HOW MANY TIMES to multiply the base by itself. 2³ = 8.
Ordering negatives backward	−2 is greater than −5, but students often think larger absolute value = larger number.	Draw a quick number line. Farther left = smaller.
Mixing up GCF and LCM	Students use the lowest powers when they need the highest, or vice versa.	GCF = Greatest → shared factors → pick lowest powers. LCM = Least → all primes → pick highest powers.
Forgetting that any number to the 0 power is 1	Students think n⁰ = 0 or n⁰ = n.	Memorize: b⁰ = 1 for any nonzero b. This is on the GED formula reference.

Watch out for these common pitfalls on the GED

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KEY TAKEAWAY

When you see an exponent like 5³, think of it as a shorthand recipe: "Start with 5, and multiply it by itself 3 total times." It's like telling a copier to make 3 copies of the same page — the exponent is the number of copies, and the base is what you're copying.

SECTION 8

Connecting to Advanced Topics

The skills you've learned in this lesson are stepping stones to more advanced GED topics. Understanding factors and multiples directly feeds into working with fractions (simplifying and finding common denominators). Exponents lead into scientific notation, which the GED tests when dealing with very large or very small numbers. The table below shows how today's concepts connect to what comes next.

Concept from This Lesson	Where It Leads on the GED
Ordering numbers	Comparing values in data sets, inequalities, and number line problems
Finding factors & GCF	Simplifying fractions, factoring algebraic expressions
Finding multiples & LCM	Adding fractions with unlike denominators, solving scheduling/ratio problems
Exponents (positive)	Scientific notation, area and volume formulas, compound interest
Exponent rules (product, power)	Simplifying algebraic expressions with variables and exponents

On the GED, about 55% of questions focus on algebraic problem solving, where exponent rules are especially important. The remaining 45% focus on quantitative problem solving, where ordering, factors, and multiples appear in practical contexts like comparing measurements, splitting quantities, and interpreting data. Mastering this lesson sets you up for success across both strands.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A student says, "Since 8 is a factor of 24, that means 24 is a multiple of 8." Which of the following best describes this statement?

A. The statement is incorrect because factors and multiples are unrelated concepts.
B. The statement is correct because if b is a factor of a, then a is always a multiple of b.
C. The statement is incorrect because 24 is a factor of 8, not a multiple of 8.
D. The statement is correct only when both numbers are even.

Show answer (B)

The correct answer is B. Factors and multiples describe the same relationship from opposite perspectives. Since 24 ÷ 8 = 3 with no remainder, 8 is a factor of 24 AND 24 is a multiple of 8. Choice A is wrong because factors and multiples are directly related. Choice C reverses the relationship — 24 is larger, so it cannot be a factor of 8. Choice D is wrong because this relationship holds for any whole numbers, not just even ones.

Question 2: BASIC CALCULATION

A store displays the following sale prices: $3.50, $0.99, $3.05, $3.55. Which list shows these prices in order from least to greatest?

A. $0.99, $3.05, $3.50, $3.55
B. $0.99, $3.05, $3.55, $3.50
C. $3.55, $3.50, $3.05, $0.99
D. $0.99, $3.50, $3.05, $3.55

Show answer (A)

The correct answer is A. Start by identifying the smallest value: $0.99 is clearly less than the others since it's under $1. Among the remaining three, compare digits left to right: all start with 3, so look at the cents. $3.05 < $3.50 < $3.55. Choice B swaps $3.50 and $3.55, which is incorrect since 50 cents < 55 cents. Choice C is in descending order (greatest to least), which is the opposite of what was asked. Choice D places $3.50 before $3.05, which is incorrect.

Question 3: INTERMEDIATE

Two buses leave the same station at 6:00 AM. Bus A returns every 12 minutes and Bus B returns every 18 minutes. At what time will both buses next be at the station together?

A. 6:24 AM
B. 6:30 AM
C. 6:36 AM
D. 6:48 AM

Show answer (C)

The correct answer is C. This is an LCM problem: we need the smallest number of minutes divisible by both 12 and 18. Find the prime factorizations: 12 = 2² × 3 and 18 = 2 × 3². The LCM takes the highest power of each prime: 2² × 3² = 4 × 9 = 36. So both buses return together after 36 minutes, at 6:36 AM. Choice A (24 minutes) is a multiple of 12 but not 18. Choice B (30 minutes) is not a multiple of either. Choice D (48 minutes) is a multiple of 12 but not 18.

Question 4: APPLIED

A bacteria colony doubles every hour. If the colony starts with 500 bacteria at noon, a scientist models the population after h hours as 500 × 2ʰ. How many bacteria are in the colony at 5:00 PM on the same day?

A. 5,000
B. 10,000
C. 16,000
D. 32,000

Show answer (C)

The correct answer is C. From noon to 5:00 PM is 5 hours, so h = 5. Substitute: 500 × 2⁵ = 500 × 32 = 16,000 bacteria. Choice A (5,000) likely comes from multiplying 500 × 10 or 500 × 2 × 5, which misapplies the exponent. Choice B (10,000) might result from computing 500 × 20, confusing exponential and linear growth. Choice D (32,000) results from using 2⁶ = 64 instead of 2⁵ = 32, miscounting the hours.

Question 5: CRITICAL THINKING

3 pts

A warehouse manager needs to pack 60 small boxes and 84 large boxes into shipping crates. Each crate must contain the same number of small boxes and the same number of large boxes, with no boxes left over. Part A: What is the greatest number of crates the manager can fill? Part B: How many small boxes and how many large boxes will be in each crate? Show your reasoning.

Show answer

Rubric (3 points total): Point 1 — Correctly identifies this as a GCF problem and finds the prime factorizations: 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7. Point 2 — Correctly computes the GCF by taking the lowest power of each shared prime: GCF = 2² × 3 = 12. States the manager can fill 12 crates. Point 3 — Correctly computes the contents of each crate: 60 ÷ 12 = 5 small boxes per crate and 84 ÷ 12 = 7 large boxes per crate. A complete answer states: 12 crates, each containing 5 small boxes and 7 large boxes.

SUMMARY

Lesson Summary

Summary

In this lesson, you learned four foundational number skills tested on the GED. Ordering numbers means arranging values from least to greatest or greatest to least — convert fractions to decimals when comparing mixed types, and remember that negative numbers decrease as they move farther from zero. Factors are numbers that divide evenly into a given number, and the Greatest Common Factor (GCF) is found by taking the lowest power of each shared prime factor. Multiples are the products you get by multiplying a number by 1, 2, 3, and so on, and the Least Common Multiple (LCM) uses the highest power of every prime factor involved.

Exponents represent repeated multiplication: bⁿ means multiplying the base b by itself n times. Key rules to remember: any nonzero number to the 0 power equals 1, multiplying same bases means adding exponents, and raising a power to a power means multiplying exponents. Memorize common powers of 2, 3, 5, and 10 for the no-calculator section. These skills are the building blocks for fractions, algebra, scientific notation, and nearly every other topic on the GED Mathematical Reasoning test.