Curriculum Core Rules & Strategy
Always write out given values and the relevant formula before substituting numbers.
Substitute your final answer back into the original problem to verify correctness.
Double-check negative signs, common denominators, and arithmetic order of operations.
Mastering GCF & LCM with SolveCalc
GCF (Greatest Common Factor) finds the largest shared factor — used to simplify fractions. LCM (Least Common Multiple) finds the smallest shared multiple — used for common denominators. They are linked by: GCF(a,b) × LCM(a,b) = a × b.
GCF — The Largest Shared Factor
The GCF (also called HCF — Highest Common Factor) of two numbers is the largest factor that divides both evenly. Three methods to find it:
- Listing factors: List all factors of each number; pick the largest in common.
- Prime factorization: Factor each number; multiply the common prime factors (using the lower exponent).
- Euclidean Algorithm: GCF(a,b) = GCF(b, a mod b). Repeat until remainder is 0.
LCM — The Smallest Shared Multiple
The LCM of two numbers is the smallest number that is a multiple of both. Three methods:
- Listing multiples: List multiples of each; find the first one in common.
- Prime factorization: Factor each number; take the highest power of every prime that appears.
- GCF relationship: LCM(a,b) = (a × b) ÷ GCF(a,b).
The Fastest GCF Method — Euclidean Algorithm
For large numbers, the Euclidean Algorithm is far faster than listing factors:
- Divide the larger number by the smaller and note the remainder
- Replace the larger number with the smaller, and the smaller with the remainder
- Repeat until the remainder is 0; the last non-zero divisor is the GCF
252 = 2×105 + 42
105 = 2×42 + 21
42 = 2×21 + 0
GCF = 21
GCF & LCM Problems — Solved
GCF(36,48)=12. 36÷12=3, 48÷12=4. Answer: 3/4
12=2²×3, 18=2×3², 30=2×3×5. LCM=2²×3²×5 = 180
LCM(36,48)=144s = 2 minutes 24 seconds
SolveCalc Insight
DEEP DIVESolveCalc provides free, browser-based math tools for students and educators at every level. Use our interactive calculators alongside this curriculum guide to verify your calculations and explore concepts hands-on — no sign-up, no downloads, no limits.
Explore with SolveCalc's Math Tools Library→Frequently Asked Questions
What is the GCF of 24 and 36? +
GCF(24, 36) = 12. Both 24 and 36 are divisible by 12, and no number larger than 12 divides both evenly.
What is the LCM of 4 and 6? +
LCM(4, 6) = 12. 12 is the smallest number that is a multiple of both 4 (4×3=12) and 6 (6×2=12).
What is the relationship between GCF and LCM? +
GCF(a,b) × LCM(a,b) = a × b. This formula lets you find the LCM once you know the GCF.