2.2 Prime Factors, Divisibility, GCF & LCM

Prime Factors, Divisibility, GCF & LCM

Understanding the structural composition of whole numbers through prime factors, divisibility tests, and common multiples is a core strand of the Ontario Grades 4–8 mathematics curriculum. EQAO's published Number Sense skills name operations with whole numbers, integers, decimals and fractions rather than factorization as such, but GCF and LCM are the machinery those operations run on: you cannot add unlike fractions without a common multiple, and you cannot reduce a result to lowest terms without a common factor. Expect these techniques to appear inside fraction, ratio, and proportional-reasoning items rather than as standalone "find the GCF" questions.

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Prime vs. Composite Numbers

• Prime Number: A positive integer strictly greater than $1$ that has exactly two distinct positive factors: $1$ and itself.
• Composite Number: A positive integer strictly greater than $1$ that has more than two factors (it can be divided evenly by numbers other than $1$ and itself).

[!WARNING] Special Cases: 0 and 1

• $1$ is NEITHER prime nor composite. It has only one positive factor ($1$).
• $0$ is NEITHER prime nor composite. It has infinitely many factors because any non-zero number divides $0$ ($0 \div n = 0$).
• $2$ is the ONLY EVEN PRIME NUMBER. All other even numbers are composite because they have $2$ as a factor.

Primes Under 50

There are 15 prime numbers less than 50: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$

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Ontario Curriculum Divisibility Rules

Divisibility rules allow teacher candidates to quickly determine if an integer is divisible by another without performing full long division.

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Prime Factorization & Factor Trees

The Fundamental Theorem of Arithmetic

Every integer greater than $1$ is either a prime number itself or can be represented as a unique product of prime numbers (up to the order of factors).

Constructing a Factor Tree

To find the prime factorization of a number (e.g., $360$):

1. Split $360$ into any two factors: $360 = 18 \times 20$.
2. Split $18$ into $2 \times 9$, and $20$ into $4 \times 5$.
3. Continue splitting composite factors until all branch ends are prime numbers:
   • $9 = 3 \times 3$
   • $4 = 2 \times 2$
4. Collect all prime ends: $2, 2, 2, 3, 3, 5$.
5. Write in Standard Prime Index Form:
   $360 = 2^3 \times 3^2 \times 5^1$

          360
         /   \
       18  x  20
      /  \   /  \
     2 x  9 4  x 5
         /\ /\
        3 3 2 2

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GCF & LCM Algorithms

1. Greatest Common Factor (GCF)

The largest positive integer that divides each of two or more integers without a remainder.

• Prime Factor Rule: Express each number in prime index form. The GCF is the product of the lowest power of each common prime factor.

2. Least Common Multiple (LCM)

The smallest positive integer that is divisible by each of two or more integers.

• Prime Factor Rule: Express each number in prime index form. The LCM is the product of the highest power of every prime factor present across all numbers.

Fundamental Product Identity

For any two positive integers $a$ and $b$: $\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b$

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Contextual Application Matrix: Word Problem Strategy

When faced with MPT contextual word problems, use this quick reference table to determine whether to solve for GCF or LCM:

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Detailed Step-by-Step Worked Examples

Example 2.2.1: Factor Tree & Prime Index Form

Construct the prime factorization of $504$ using the factor tree method and express the answer in standard prime index form.

Solution Step-by-Step:

1. First Factor Pair: Notice $504$ is even. Divide by $2$:
   $504 = 2 \times 252$

2. Decompose $252$: Sum of digits is $2+5+2=9$, so it is divisible by $9$:
   $252 = 9 \times 28$

3. Decompose Prime Branches:

   • $9 = 3 \times 3$
   • $28 = 4 \times 7 = 2 \times 2 \times 7$

4. Gather All Prime Terminals:
   Primes are: $2, 3, 3, 2, 2, 7$.

5. Group and Express with Exponents:

   • Count of $2$s: $3 \longrightarrow 2^3$
   • Count of $3$s: $2 \longrightarrow 3^2$
   • Count of $7$s: $1 \longrightarrow 7^1$
     $504 = 2^3 \times 3^2 \times 7^1$

Final Answer: $2^3 \times 3^2 \times 7^1$

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Example 2.2.2: GCF and LCM of Three Numbers

Determine the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of $72$, $108$, and $120$.

Solution Step-by-Step:

1. Write Prime Factorizations:

   • $72 = 8 \times 9 = 2^3 \times 3^2$
   • $108 = 4 \times 27 = 2^2 \times 3^3$
   • $120 = 8 \times 15 = 2^3 \times 3^1 \times 5^1$

2. Calculate GCF (Product of Lowest Powers of COMMON Factors):

   • Common prime factors present in ALL three numbers are $2$ and $3$.
   • Lowest power of base $2$: $\min(3, 2, 3) = 2^2$
   • Lowest power of base $3$: $\min(2, 3, 1) = 3^1$ $\text{GCF} = 2^2 \times 3^1 = 4 \times 3 = 12$

3. Calculate LCM (Product of Highest Powers of ALL Factors):

   • All prime factors present: $2, 3, 5$.
   • Highest power of base $2$: $\max(3, 2, 3) = 2^3$
   • Highest power of base $3$: $\max(2, 3, 1) = 3^3$
   • Highest power of base $5$: $\max(0, 0, 1) = 5^1$ $\text{LCM} = 2^3 \times 3^3 \times 5^1 = 8 \times 27 \times 5 = 1,080$

Final Answer: $\text{GCF} = 12$; $\text{LCM} = 1,080$

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Example 2.2.3: Integrated Dual MPT Word Problem

An Ontario school coordinator has $96$ blue markers, $144$ red markers, and $120$ green markers.

1. What is the maximum number of identical activity kits she can assemble such that all markers are distributed evenly with no remainders, and how many red markers are in each kit?
2. If two automated school bells ring every $18$ minutes and $24$ minutes respectively, and they ring simultaneously at 8:00 AM, at what time will they next ring together?

Solution Step-by-Step:

Part 1: Kit Distribution (GCF Problem)

1. Prime Factorizations:
   • $96 = 2^5 \times 3^1$
   • $144 = 2^4 \times 3^2$
   • $120 = 2^3 \times 3^1 \times 5^1$
2. $\text{GCF} = 2^{\min(5,4,3)} \times 3^{\min(1,2,1)} = 2^3 \times 3^1 = 8 \times 3 = 24$ kits.
3. Red markers per kit: $\frac{144 \text{ total red}}{24 \text{ kits}} = 6$ red markers per kit.

Part 2: Bell Overlap (LCM Problem)

1. Prime Factorizations:
   • $18 = 2^1 \times 3^2$
   • $24 = 2^3 \times 3^1$
2. $\text{LCM} = 2^{\max(1,3)} \times 3^{\max(2,1)} = 2^3 \times 3^2 = 8 \times 9 = 72$ minutes.
3. Convert $72$ minutes to hours and minutes: $72 \text{ min} = 1 \text{ hour } 12 \text{ minutes}$.
4. Add to 8:00 AM: $8:00 \text{ AM} + 1 \text{h } 12 \text{m} = 9:12 \text{ AM}$.

Final Answer: Max kits = $24$ (with $6$ red markers per kit); Next simultaneous chime = 9:12 AM.