GCF/LCM fluency sprint

Rapidly determine GCF and LCM of small integers.

A fluency sprint: build the speed and accuracy you need to do this calculation instantly on test day.

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How to find the GCF and LCM fast

Factor both numbers into primes.

Quick LCM for small numbers: list multiples of the larger number until the smaller one divides it. For 12 and 18: \(18, 36\), and 12 divides 36.

Useful rule: GCF \(\times\) LCM \(=\) the product of the two numbers. So the LCM of 24 and 20 is \(480 \div 4 = 120\).

Which one to use: "next time together" (buses, lights) is an LCM; "the largest equal groups" is a GCF. To combine ratios \(a:b\) and \(b:c\), scale both so that \(b\) becomes the LCM of its two values.

Common slip: using the product (\(24 \times 20 = 480\)) instead of the LCM. It is a common multiple, but not the least one.

Worked examples

Example 1.
Bus A leaves the station every 6 minutes, and Bus B leaves every 18 minutes. Both leave at 7:45 a.m. At what time do they next leave at the same time?
Show answer and explanation
Answer:
D

They leave together after a common multiple of \(6\) and \(18\); the next time is the LEAST common multiple, \(18\) minutes. 7:45 a.m. \(+ 18\) minutes \(=\) 8:03 a.m. (Using the product \(6 \times 18 = 108\) minutes gives 9:33 a.m. Using the GCF, \(6\) minutes, instead of the LCM gives 7:51 a.m. Adding the two gaps (\(6 + 18 = 24\) minutes) gives 8:09 a.m.)

Example 2.
What is the greatest common factor (GCF) of 50 and 60?
Show answer and explanation
Answer:
C

Factor both numbers: \(50 = 2 \times 5^{2}\) and \(60 = 2^{2} \times 3 \times 5\). The GCF takes the smaller power of each shared prime: GCF(50, 60) = 10. Finding the LCM instead of the GCF gives \(300\). Stopping at the first common factor, 2 gives \(2\). Taking the larger power of each shared prime gives \(100\).

Example 3.
In a group of students, the ratio of first graders to second graders is 8:7, and the ratio of second graders to third graders is 4:3. What is the combined ratio of first graders to second graders to third graders, in simplest form?
Show answer and explanation
Answer:
D

The shared term is second graders: scale 8:7 by 4 and 4:3 by 7 so both give second graders \(= 28\) (the LCM of 7 and 4). This gives 32:28:21, which is already in lowest terms. (Joining the original numbers without matching second graders gives 8:7:3 or 8:4:3. Writing 28 for second graders but swapping the factors on the outer terms (\(8 \times 7\) and \(3 \times 4\)) gives 56:28:12, which reduces to 14:7:3.)