Prime factorization is a compact inventory of a number's building blocks. The GCF keeps only factors shared by every expression at their smallest common powers; the LCM includes every required factor at its greatest power.
Learning objectives
- Distinguish prime, composite, and the special number \(1\).
- Create and validate a factor tree.
- Write numeric and monomial prime factorizations.
- Find GCFs using minimum shared exponents.
- Find LCMs using maximum required exponents.
Prime numbers and prime factorization
- Prime number
- A positive integer greater than \(1\) with exactly two positive factors: \(1\) and itself. The number \(1\) is neither prime nor composite.
Prime factor tree for 504
Factor tree decomposing 504 into prime factors; every pair of children multiplies to its parent.
\[504=2^{3}\cdot3^{2}\cdot7\]
GCF and LCM from exponent inventories
| Question | GCF | LCM |
|---|---|---|
| Which prime bases? | Only bases shared by all expressions | Every base appearing in any expression |
| Which exponent? | Smallest shared exponent | Largest required exponent |
| Meaning | Greatest expression dividing each input | Least positive expression divisible by each input |
| For \(12=2^2\cdot3\), \(18=2\cdot3^2\) | \(2^1\cdot3^1=6\) | \(2^2\cdot3^2=36\) |
Find monomial GCF and LCM
Find the GCF and LCM of \(18x^4y^2\) and \(30x^2y^5\).
- Coefficients
\(18=2\cdot3^2\) and \(30=2\cdot3\cdot5\), so numeric GCF is \(6\) and LCM is \(90\).
- Variables
Use minima \(x^2y^2\) for GCF and maxima \(x^4y^5\) for LCM.
Choose minimum and maximum powers
For \(24a^3b^2\) and \(36a^2b^5\), what variable part belongs in the GCF and LCM?
Show answer and explanation
Answer: GCF variable part \(a^2b^2\); LCM variable part \(a^3b^5\).
Use the smaller exponent for each shared base in the GCF and the larger exponent in the LCM.
Key takeaways
What to remember
- \(1\) is neither prime nor composite.
- A complete factor tree has prime leaves.
- GCF uses shared minimum powers.
- LCM uses all maximum powers.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.