Find greatest common factor and least common multiple of integers.
Free, timed SHSAT practice sets on gcf & lcm— every question includes a full worked explanation, and there's no account required.
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Write each number as a product of primes first. Then:
Example with 20 and 50: \(20 = 2^{2} \times 5\) and \(50 = 2 \times 5^{2}\). GCF \(= 2 \times 5 = 10\). LCM \(= 2^{2} \times 5^{2} = 100\).
A useful check: GCF \(\times\) LCM \(=\) the product of the two numbers. Here \(10 \times 100 = 1000 = 20 \times 50\).
Quick facts:
Common traps: mixing up GCF and LCM, and using the product as the LCM when the numbers share a factor (8 and 10 give 40, not 80).
Factor both numbers: \(64 = 2^{6}\) and \(36 = 2^{2} \times 3^{2}\). The GCF takes the smaller power of each shared prime: GCF(64, 36) = 4. Finding the LCM instead of the GCF gives \(576\). Stopping at the first common factor, 2 gives \(2\). Taking the larger power of each shared prime gives \(64\).
LCM(12, 14) = \((12 \times 14) \div \text{GCF}\) = \((12 \times 14) \div 2 = 84\). Finding the GCF instead of the LCM gives \(2\). Multiplying 12 and 14 without dividing out the GCF gives \(168\). Using only the primes both numbers share gives \(4\).
LCM(18, 30) = \((18 \times 30) \div \text{GCF}\) = \((18 \times 30) \div 6 = 90\). Finding the GCF instead of the LCM gives \(6\). Multiplying 18 and 30 without dividing out the GCF gives \(540\). Using only the primes both numbers share gives \(18\).