Factors and Fractions

Prime Factorization, GCF and LCM

Find GCF and LCM using prime factorization. The GCF is the product of common primes at their lowest powers; the LCM uses highest powers.

prime-factorization-gcf-lcm To find GCF and LCM by prime factorization: - Factor each number into primes. - GCF: multiply common primes at lowest exponent. - LCM: multiply all primes at highest exponent.

Example

Use prime powers to find the GCF and LCM of 24 and 36.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

243624 \quad 36
Break 24 and 36 into prime chips, then compare the chips.24first number36second numberPrime chips make the GCF and LCM choices visible.

Step 2 — Factor first number

Use repeated division to factor 24.

24=23×324 = \hlmath{2^{3}} \times \hl{3}
The leaves are all prime factors of 24.24222324 = 2 x 2 x 2 x 3

Step 3 — Factor second number

Use repeated division to factor 36.

36=22×3236 = \hlmath{2^{2}} \times \hlmath{3^{2}}
The leaves are all prime factors of 36.36223336 = 2 x 2 x 3 x 3

Step 4 — Common lowest powers

For the GCF, use common primes with lowest powers.

GCF terms: 22×3\mathrm{GCF\ terms:}\ \hlmath{2^{2}} \times \hl{3}
Circle only the prime chips both numbers share.2436222322332 x 2 x 3 = 12Blue chips show the lowest shared count of each prime.

Step 5 — GCF

The greatest common factor is 12.

GCF=12\mathrm{GCF}= \hl{12}
Shared chips multiply to the GCF: 12.2436222322332 x 2 x 3 = 12Blue chips show the lowest shared count of each prime.

Step 6 — Highest powers

For the LCM, use every prime with highest powers.

LCM terms: 23×32\mathrm{LCM\ terms:}\ \hlmath{2^{3}} \times \hlmath{3^{2}}
Cover every prime count needed by either number.243622232233Use the higher count for each prime.222332 x 2 x 2 x 3 x 3 = 72

Step 7 — LCM

The least common multiple is 72.

LCM=72\mathrm{LCM}= \hl{72}
Covered chips multiply to the LCM: 72.243622232233Use the higher count for each prime.222332 x 2 x 2 x 3 x 3 = 72

Step 8 — Result

Read the final result.

127212 \quad 72
Read the overlap and coverage results.GCF: 2 x 2 x 312LCM: 2 x 2 x 2 x 3 x 372Overlap gives the GCF; full coverage gives the LCM.