5.5 Factors, Multiples, Primes, GCF & LCM

5.5 Factors, Multiples, Primes, GCF & LCM

Quick Answer: A factor of a number divides it evenly; a multiple is what you get when you multiply it by a whole number. A prime has exactly two distinct factors (1 and itself); a composite has more than two; 1 is neither. Prime factorization drives everything else: the greatest common factor (GCF) is the product of shared primes at their lowest powers, and the least common multiple (LCM) is the product of all primes at their highest powers. In word problems, GCF splits things into the largest equal groups and LCM finds when two repeating cycles line up.

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Where This Sits in the Blueprint

Factor and multiple reasoning runs across several grades and reporting categories:

That grade 6 benchmark is the one students most often lose points on, because it does not stop at finding the GCF - it asks them to use it to rewrite an expression.

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Factors and Multiples: Opposite Directions

The two words point in opposite directions from the same number, and mixing them up is the single most common error in this topic.

                    <-- FACTORS                 MULTIPLES -->
                 1, 2, 3, 4, 6, 12   [ 12 ]   12, 24, 36, 48, 60, ...
                 (finite - they fit inside)   (infinite - they extend outward)

• Factors of 12: 1, 2, 3, 4, 6, 12. There are six of them, and there will never be a seventh.
• Multiples of 12: 12, 24, 36, 48, 60, ... The list never ends.

Finding factor pairs systematically. Do not hunt randomly. Start at 1 and walk upward, recording pairs, until the pairs cross:

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Nine factors. Stopping at the square root ($6 \times 6$) guarantees you have them all and saves half the work.

Divisibility Shortcuts Worth Memorizing

Applied to 4,518: the digit sum is $4+5+1+8 = 18$, which is divisible by both 3 and 9, and the last digit is even - so 4,518 is divisible by 2, 3, 6, and 9.

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Prime, Composite, and the Number 1

• Prime: exactly two distinct factors, 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
• Composite: more than two factors. 4, 6, 8, 9, 10, 12, ...
• 1 is neither. It has exactly one factor. This is not a technicality invented to trick students - it is why prime factorization is unique.
• 2 is the only even prime. Every other even number has 2 as a third factor.

[!IMPORTANT] Test items love 1, 2, and 9. 1 is neither prime nor composite. 2 is prime even though it is even. 9 is composite even though it is odd ($9 = 3 \times 3$). Odd does not mean prime.

Prime Factorization

Break a number down until only primes remain. A factor tree is the standard tool:

        72                         180
       /  \                       /   \
      8    9                    18     10
     / \  / \                  /  \   /  \
    2   4 3  3                 2    9 2    5
       / \                         / \
      2   2                       3   3

  72 = 2 x 2 x 2 x 3 x 3        180 = 2 x 2 x 3 x 3 x 5
     = 2^3 x 3^2                    = 2^2 x 3^2 x 5

The starting split does not matter. Beginning $72 = 2 \times 36$ instead of $72 = 8 \times 9$ produces the same primes. That uniqueness is what makes the method trustworthy.

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GCF and LCM from Prime Factorization

Once both numbers are in prime-power form, the rules are mechanical.

The 5 appears only in 180, so it is excluded from the GCF and included in the LCM. That single decision is where most errors happen.

The Check That Catches Almost Every Mistake

For any two positive integers:

$\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b$

Check the example: $36 \times 360 = 12{,}960$, and $72 \times 180 = 12{,}960$. They match, so both answers are right. If they do not match, one of the two is wrong - and it is almost always the GCF, from including a prime that only one number has.

Using the GCF with the Distributive Property

Grade 6 asks students to rewrite a sum by pulling out the GCF. Take $48 + 36$:

• $48 = 2^4 \times 3$ and $36 = 2^2 \times 3^2$, so the GCF is $2^2 \times 3 = 12$.
• $48 = 12 \times 4$ and $36 = 12 \times 3$.
• Therefore $48 + 36 = 12(4 + 3)$.

Verify: $12 \times 7 = 84$, and $48 + 36 = 84$. The parenthesized numbers must be relatively prime - here 4 and 3 share no factor but 1. If they still share a factor, you pulled out a common factor but not the greatest one.

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Deciding Between GCF and LCM in Word Problems

This is the actual skill. The arithmetic is easy; choosing the right tool is not.

GCF problem. A teacher has 48 pencils and 36 erasers and wants identical kits with nothing left over. The most kits possible is $\text{GCF}(48,36) = 12$ kits, each holding 4 pencils and 3 erasers.

LCM problem. One bus leaves every 12 minutes and another every 18 minutes, both departing at 8:00 a.m. $18 = 2 \times 3^2$ and $12 = 2^2 \times 3$, so the LCM is $2^2 \times 3^2 = 36$. They leave together again at 8:36 a.m.

[!IMPORTANT] Sanity-check the direction. A GCF answer is never larger than the smaller given number. An LCM answer is never smaller than the larger given number. If your bus answer came out as 6 minutes, you computed a GCF for an LCM question.

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Common Exam Traps & Misconceptions

[!WARNING]

Trap 1: Calling 1 prime

One is neither prime nor composite. Items testing this usually appear as "select all the prime numbers" with 1 sitting in the list looking harmless.

[!WARNING]

Trap 2: Finding a common factor instead of the greatest one

For 48 and 36, 6 is a common factor and 12 is the greatest. If a rewritten expression still has a shared factor inside the parentheses - $6(8+6)$ rather than $12(4+3)$ - the factoring is incomplete.

[!WARNING]

Trap 3: Reversing GCF and LCM in a word problem

"Greatest" in the question does not automatically mean GCF, and "least" does not automatically mean LCM. Ask what the number is: the size of a group, or a point in time. Groups mean GCF; coincidences mean LCM.

[!WARNING]

Trap 4: Skipping a prime that appears in only one number

Building an LCM from shared primes only is the classic error. The LCM must be divisible by both numbers, so every prime in either factorization has to appear.