Apply Number Properties — Apply factors, multiples, prime factorization, GCD, and LCM. - Praxis Core Math

SECTION 1

Historical Context & Motivation

The concepts of factors, multiples, and prime numbers are among the oldest and most enduring ideas in mathematics. Long before algebra or calculus existed, ancient civilizations grappled with questions of divisibility: How can a harvest be divided equally among workers? When do two periodic events coincide? These practical demands drove the development of a rich theory of number properties that remains central to mathematics education today. As a future educator preparing for the PRAXIS Core, understanding the deep roots and logical architecture of these concepts will strengthen both your test performance and your pedagogical clarity.

c. 1800 BCE • Babylonian Tables of Reciprocals

Babylonian scribes created tables listing factor pairs and reciprocals in base-60, enabling efficient division for trade, land allocation, and astronomical computation. Their sexagesimal system (base 60) was chosen partly because 60 has many factors — 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 — making division convenient.

c. 300 BCE • Euclid's Elements — The Euclidean Algorithm

In Books VII–IX of the Elements, Euclid formalized the algorithm for finding the greatest common divisor (GCD) of two numbers using repeated subtraction. He also proved that the set of prime numbers is infinite, establishing a cornerstone of number theory.

c. 240 BCE • Sieve of Eratosthenes

The Greek mathematician Eratosthenes devised a systematic method for listing all prime numbers up to any given limit by iteratively eliminating multiples of each prime. This sieve remains a standard algorithmic tool taught at every level of mathematics education.

1801 • Gauss's Disquisitiones Arithmeticae

Carl Friedrich Gauss published a comprehensive treatise that formalized the Fundamental Theorem of Arithmetic — every integer greater than 1 has a unique prime factorization. This result provides the theoretical foundation for GCD and LCM computations via prime factorization.

Modern Era • Curriculum Standards & Certification Exams

Contemporary standards (CCSS, NCTM) place factors, multiples, and prime factorization in the upper-elementary and middle-school curriculum. The PRAXIS Core (5733) tests prospective teachers' fluency with these tools, reflecting their importance across fraction operations, ratio reasoning, and algebraic thinking.

The central question these ideas address is deceptively simple: What is the internal multiplicative structure of a whole number, and how can we use that structure to relate two or more numbers? Mastering the answer equips you to simplify fractions, find common denominators, solve word problems involving cycles and grouping, and — crucially — to explain these processes to your own students with precision and confidence.

SECTION 2

Core Principles & Definitions

Before diving into procedures, it is essential to anchor the vocabulary with precise definitions. The five key concepts in this lesson — factors, multiples, prime factorization, greatest common divisor, and least common multiple — form an interlocking system. Each concept depends on and illuminates the others, so a clear grasp of terminology will streamline every computation that follows.

Factors (Divisors)

A factor of a positive integer n is any positive integer that divides n without a remainder. Equivalently, if n = a × b, then both a and b are factors of n.

Multiples

A multiple of n is any product n × k where k is a positive integer. Factors look inward (what divides n); multiples look outward (what n divides into).

Prime Factorization

Every integer greater than 1 can be expressed as a product of prime numbers in exactly one way (up to ordering). This unique decomposition is guaranteed by the Fundamental Theorem of Arithmetic and serves as the 'DNA' of a number.

Greatest Common Divisor (GCD)

The GCD of two integers is the largest positive integer that divides both. It is found by taking the lowest power of each shared prime factor from their factorizations, or by applying the Euclidean algorithm.

Least Common Multiple (LCM)

The LCM of two integers is the smallest positive integer that is a multiple of both. It is found by taking the highest power of every prime factor appearing in either factorization.

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KEY TAKEAWAY

Think of the prime factorization of a number as its molecular formula. Just as water is always H₂O, the number 60 is always 2² × 3 × 5. The GCD is like the shared molecular fragment between two compounds — the largest substructure they have in common. The LCM is like the smallest molecule that contains both compounds as sub-units. Once you internalize this 'chemistry of numbers,' computing GCDs and LCMs becomes a systematic comparison of prime ingredients rather than a trial-and-error search.

SECTION 3

Visual Explanation — Factor Trees & Prime Factorization

A factor tree is the standard visual tool for decomposing a composite number into its prime factors. Starting from the original number at the top, each branch splits a composite into two factors; the process continues until every leaf is a prime. The diagram below illustrates factor trees for 60 and 84, the two numbers we will use throughout this lesson to demonstrate GCD and LCM calculations.

Each factor tree decomposes a composite number by repeatedly splitting it into two factors until only prime leaves remain. The blue nodes represent composite intermediaries, while the purple nodes represent primes. Collecting all the prime leaves yields the unique prime factorization: 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7.

Notice that the path you choose through the factor tree does not affect the final result. You could start by splitting 60 into 4 × 15 instead of 2 × 30, and the prime leaves would still be {2, 2, 3, 5}. This invariance is precisely what the Fundamental Theorem of Arithmetic guarantees: the prime factorization is unique regardless of the decomposition order. Grasping this fact is pedagogically critical because students sometimes worry that different factor tree paths yield different answers. You can reassure them with confidence that the destination is always the same.

SECTION 4

Mathematical Framework

The procedural backbone of this topic rests on four interconnected formulas and methods. Each builds on the notion of prime factorization, transforming what might otherwise be guesswork into an algorithmic, teachable procedure. Understanding these formulas at a structural level will help you not only solve PRAXIS items efficiently but also articulate the reasoning to students.

FUNDAMENTAL THEOREM OF ARITHMETIC

n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ

Every integer n > 1 can be written uniquely as a product of primes p₁ < p₂ < … < pₖ raised to positive integer exponents a₁, a₂, …, aₖ. This unique representation is the foundation for computing GCD and LCM.

GCD VIA PRIME FACTORIZATION

GCD(a, b) = p₁^min(α₁,β₁) × p₂^min(α₂,β₂) × … × pₖ^min(αₖ,βₖ)

If a = ∏ pᵢ^αᵢ and b = ∏ pᵢ^βᵢ (using 0 for missing primes), the GCD takes the minimum exponent for each prime. Only shared primes contribute.

LCM VIA PRIME FACTORIZATION

LCM(a, b) = p₁^max(α₁,β₁) × p₂^max(α₂,β₂) × … × pₖ^max(αₖ,βₖ)

The LCM takes the maximum exponent for each prime factor. Every prime that appears in either factorization is included.

GCD–LCM PRODUCT IDENTITY

GCD(a, b) × LCM(a, b) = a × b

This elegant identity provides a shortcut: once you know the GCD, compute LCM = (a × b) ÷ GCD(a, b), and vice versa. It also serves as a powerful verification check.

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The Euclidean Algorithm — A Complementary Method

While prime factorization is intuitive and visual, the Euclidean algorithm computes the GCD without factoring. It works by repeated division: GCD(a, b) = GCD(b, a mod b) until the remainder is 0. For the PRAXIS, both methods are valid; choose whichever is faster for the given numbers.

SECTION 5

Detailed Breakdown — Comparing GCD and LCM Visually

One of the most effective ways to solidify the distinction between GCD and LCM is to align the prime factorizations side by side and examine which exponents are selected for each operation. The Venn-diagram approach — showing the shared and unshared prime factors — is widely used in classroom instruction and is particularly helpful for visual learners. The diagram below uses this approach for our running example of 60 and 84.

The Venn diagram partitions the prime factors of 60 and 84 into three regions. The intersection (green) contains the shared primes 2² and 3, yielding GCD = 12. The union of all regions contains every prime factor at its highest power (2², 3, 5, 7), yielding LCM = 420. The product identity confirms: 12 × 420 = 5,040 = 60 × 84.

Prime	Exponent in 60	Exponent in 84	min (→ GCD)	max (→ LCM)
2	2	2	2	2
3	1	1	1	1
5	1	0	0	1
7	0	1	0	1
Product	60	84	12	420

Prime-exponent comparison table for GCD and LCM of 60 and 84

The table above crystallizes the algorithm: list every prime that appears in either factorization, record the exponent (using 0 when a prime is absent), and then apply min for GCD or max for LCM. This systematic approach eliminates guesswork and scales effortlessly to three or more numbers — you simply extend the table with additional columns. On the PRAXIS, this tabular method is especially useful when the numbers are large enough that listing factors manually would be time-consuming.

SECTION 6

Worked Example — GCD and LCM of 120 and 90

Let us apply the full procedure to a new pair of numbers — 120 and 90 — typical of the complexity you would encounter on the PRAXIS Core. We will find the prime factorization of each, then compute the GCD and LCM, and verify using the product identity.

Find GCD(120, 90) and LCM(120, 90)

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Step 1 — Prime Factorize 120

Divide by the smallest prime repeatedly: 120 ÷ 2 = 60, 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1.

120 = 2³ × 3 × 5
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Step 2 — Prime Factorize 90

90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1.

90 = 2 × 3² × 5
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Step 3 — Identify All Primes and Their Exponents

The primes present are 2, 3, and 5. For 120: exponents are (3, 1, 1). For 90: exponents are (1, 2, 1).
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Step 4 — Compute GCD (take minimum exponents)

min(3,1) = 1 for prime 2; min(1,2) = 1 for prime 3; min(1,1) = 1 for prime 5. Thus GCD = 2¹ × 3¹ × 5¹.

GCD(120, 90) = 30
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Step 5 — Compute LCM (take maximum exponents)

max(3,1) = 3 for prime 2; max(1,2) = 2 for prime 3; max(1,1) = 1 for prime 5. Thus LCM = 2³ × 3² × 5 = 8 × 9 × 5.

LCM(120, 90) = 360
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Step 6 — Verify with the Product Identity

GCD × LCM = 30 × 360 = 10,800. Also, 120 × 90 = 10,800. The identity holds, confirming our work.

30 × 360 = 10,800 = 120 × 90 ✓

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PRAXIS Tip

If a problem gives you GCD and asks for LCM (or vice versa), skip the full factorization and use the product identity directly: LCM = (a × b) ÷ GCD. This shortcut can save valuable time on a timed exam.

SECTION 7

Comparing Methods for Finding GCD and LCM

On the PRAXIS Core — and in your future classroom — you will want to choose the most efficient method for the numbers at hand. There are three principal approaches: the prime factorization method, the Euclidean algorithm, and the listing method. Each has distinct strengths and limitations, which the table below summarizes.

Method	Best When…	Strengths	Limitations
Prime Factorization	Numbers are moderate-sized and factoring is straightforward (e.g., under 200)	Visual and systematic; yields both GCD and LCM simultaneously; reinforces number structure	Factoring large numbers can be slow; not practical for very large primes
Euclidean Algorithm	Numbers are large or you need only GCD (not LCM)	No factoring needed; very efficient; always terminates quickly	Gives GCD only — must use product identity for LCM; less intuitive for young learners
Listing Method	Numbers are small (under 30) or you want to verify other methods	Concrete and transparent; excellent for introductory instruction	Impractical for large numbers; easy to miss a factor or multiple

Comparison of three common methods for computing GCD and LCM

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PEDAGOGICAL INSIGHT

As a teacher, you will want to introduce the listing method first to build intuition, transition to factor trees and prime factorization for conceptual depth, and then present the Euclidean algorithm as an efficient shortcut for advanced students. This progression mirrors the concrete–representational–abstract (CRA) instructional framework widely endorsed by NCTM. The PRAXIS may test your ability to identify which method is most appropriate for a given scenario, so familiarity with all three is essential.

SECTION 8

Connections to the Broader Curriculum

Understanding GCD and LCM is not an end in itself; these concepts serve as essential tools across multiple domains of elementary and secondary mathematics. The PRAXIS Core values your ability to see these connections, and your future students will benefit immensely from an instructor who can seamlessly link number properties to fraction operations, ratio reasoning, and algebraic thinking.

Application Area	How GCD/LCM Is Used	Example
Simplifying Fractions	Divide numerator and denominator by GCD to reduce to lowest terms	84/120 → divide both by GCD(84,120)=12 → 7/10
Adding/Subtracting Fractions	LCM of denominators gives the least common denominator (LCD)	1/6 + 1/10: LCD = LCM(6,10) = 30, so 5/30 + 3/30 = 8/30
Word Problems (Cycles)	LCM determines when two periodic events next coincide	Bus A every 12 min, Bus B every 18 min: both depart together every LCM(12,18) = 36 min
Distributing Items Evenly	GCD determines the largest group size for equal distribution	48 apples and 36 oranges → largest identical bags: GCD(48,36)=12 bags
Algebraic Expressions	GCD of coefficients helps factor polynomial expressions	12x³ + 18x² → GCD(12,18) = 6, so 6x²(2x + 3)

Cross-curricular applications of GCD and LCM

Looking beyond the PRAXIS, these number properties connect to more advanced topics in abstract algebra (where GCD and LCM generalize to ideals in rings), cryptography (where prime factorization underpins RSA encryption), and computer science (where the Euclidean algorithm is a classic example of recursion and algorithmic efficiency). While these extensions are beyond the scope of the exam, awareness of them enriches your mathematical perspective and can inspire students who ask, 'When will I ever use this?'

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

True or false: If GCD(a, b) = 1, then the LCM(a, b) = a × b. Justify your answer.

Show answer

True. When GCD(a, b) = 1 (the numbers are coprime), the product identity gives LCM = (a × b) ÷ 1 = a × b. Intuitively, coprime numbers share no prime factors, so every prime from both numbers must appear in the LCM, making it equal to the full product. For example, GCD(8, 15) = 1, so LCM(8, 15) = 120 = 8 × 15.

Question 2: BASIC CALCULATION

Find the prime factorization of 180, and then determine GCD(180, 126) and LCM(180, 126).

Show answer

180 = 2² × 3² × 5 and 126 = 2 × 3² × 7. For GCD, take minimum exponents: 2^min(2,1) × 3^min(2,2) × 5^min(1,0) × 7^min(0,1) = 2¹ × 3² = 2 × 9 = 18. For LCM, take maximum exponents: 2² × 3² × 5 × 7 = 4 × 9 × 5 × 7 = 1,260. Verification: 18 × 1,260 = 22,680 = 180 × 126 ✓.

Question 3: INTERMEDIATE

A teacher wants to distribute 72 pencils, 108 erasers, and 54 rulers equally into gift bags so that each bag has the same number of each supply and nothing is left over. What is the greatest number of bags she can make, and how many of each item goes in each bag?

Show answer

The greatest number of bags is GCD(72, 108, 54). Factor each: 72 = 2³ × 3², 108 = 2² × 3³, 54 = 2 × 3³. Taking minimum exponents across all three: 2^min(3,2,1) × 3^min(2,3,3) = 2¹ × 3² = 18. So the teacher can make 18 bags. Each bag contains 72 ÷ 18 = 4 pencils, 108 ÷ 18 = 6 erasers, and 54 ÷ 18 = 3 rulers.

Question 4: APPLIED

Two flashing signs are synchronized at noon. Sign A flashes every 8 seconds and Sign B flashes every 14 seconds. At how many seconds after noon will they next flash at exactly the same moment? If they run for 10 minutes, how many times do they flash simultaneously (including the initial synchronization at time 0)?

Show answer

They next flash together at LCM(8, 14). Factor: 8 = 2³, 14 = 2 × 7. LCM = 2³ × 7 = 56 seconds. In 10 minutes = 600 seconds, the number of simultaneous flashes is ⌊600 ÷ 56⌋ + 1 (the +1 counts time 0). Since 600 ÷ 56 ≈ 10.71, they flash together at t = 0, 56, 112, …, 560 — that is 11 times. The key insight is that LCM gives the cycle length, and the count includes the initial event at t = 0.

Question 5: CRITICAL THINKING

Prove or disprove: For any three positive integers a, b, c, the identity GCD(a, b, c) × LCM(a, b, c) = a × b × c holds.

Show answer

This is FALSE in general. The product identity GCD × LCM = a × b holds only for two numbers. Counterexample: let a = 2, b = 2, c = 2. Then GCD(2,2,2) = 2 and LCM(2,2,2) = 2, so GCD × LCM = 4. But a × b × c = 8 ≠ 4. Another example: a = 6, b = 10, c = 15. GCD = 1, LCM = 30, so GCD × LCM = 30, while 6 × 10 × 15 = 900. The generalization fails because the pairwise interaction of shared factors among three or more numbers cannot be captured by a single product. This illustrates why mathematical generalizations must be proven, not assumed by analogy.

SUMMARY

Lesson Summary

Summary

This lesson established the interconnected framework of factors, multiples, and prime factorization — anchored in the Fundamental Theorem of Arithmetic — as the foundation for computing the greatest common divisor (GCD) and least common multiple (LCM). The GCD is found by selecting the minimum exponent for each shared prime, while the LCM is found by selecting the maximum exponent for every prime present in either factorization.

Three computational methods were compared: prime factorization (visual and systematic), the Euclidean algorithm (efficient for large numbers), and the listing method (concrete and introductory). The product identity GCD(a, b) × LCM(a, b) = a × b provides both a computation shortcut and a verification tool. These number properties underpin fraction simplification, common denominators, cycle and grouping problems, and algebraic factoring — making them indispensable tools for any educator's mathematical repertoire.