EQN-01-05

Factors, Multiples & Primes

Prime factorization and the GCF/LCM toolkit that powers fraction, ratio, and number-property questions.

~5 min read•Difficulty 2/5

A factor of n divides it exactly (factors of 12: 1, 2, 3, 4, 6, 12).
A multiple of n is $n \times$ an integer (multiples of 4: 4, 8, 12, …).
A prime has exactly two factors, 1 and itself (2, 3, 5, 7, 11, …). 2 is the only even prime.

Break a number into primes; it is the fingerprint of the number.

60 = 2 \times 2 \times 3 \times 5 = 2^{2} \times 3 \times 5

GCF (greatest common factor $) =$ product of the shared primes, lowest powers.
LCM (least common multiple $) =$ product of all primes, highest powers.

12 = 2^{2} \cdot 3, 18 = 2 \cdot 3^{2}

GCF = 2 \cdot 3 = 6 LCM = 2^{2} \cdot 3^{2} = 36

Handy check: GCF $\times$ LCM $=$ product of the two numbers $(6 \times 36 = 216 = 12 \times 18) .$

Worked examples

Example 1 (difficulty 2/5)

What is the GCF of 24 and 36?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Two buses leave together; one returns every 12 minutes, the other every 18. After how many minutes do they next leave together?

◎ Predict the answer first:

Prime-factorize first — GCF and LCM both fall straight out of the prime powers.
"Greatest common / largest that divides both" $\to$ GCF; "smallest common / next time together" $\to$ LCM.
Use GCF $\times$ LCM $=$ product of the numbers as a fast sanity check.

Swapping GCF and LCM

Why: Both involve "common".

Fix: GCF is a factor

(\leq

the numbers); LCM is a multiple

(\geq

the numbers).

Calling 1 a prime

Why: It has no factors other than itself.

Fix: A prime needs exactly two distinct factors; 1 has only one.