The problem with LCM and GCF (or GCD) is that they are too “number theoretic,” so students have a difficult time appreciating their importance. But here are two stories that associate these concepts with realistic (well, … somewhat realistic) phenomena.
The Story of the Race to the Rockies. Three Class 2-Za4 battery powered Scooters are in a “Race to the Rockies” beginning in Boston and ending in Denver. Scooter A is fast and lightweight but must replace its battery every 60 miles; Scooter B must replace its battery every 80 miles; Scooter C is slightly slower but its heavier battery can go 90 miles before being replaced. Battery replacement sites are therefore prepared at increments of 60, 80, and 90 miles along the route from Boston to Denver. In the direction of the race, how far from Boston is the very first battery-exchange site that can serve all three competitors? (Assuming, of course, that no scooter ever exchanges its battery “early.”) Since 60 = 2·2·3·5, 80 = 2·2·2·2·5, and 90 = 3·3·2·5, the answer is their LCM, or 2·2·2·2·3·3·5, which provides the answer of 720 miles.
Okay, we’re not going to model this race. But we can do this: Student A must place 8 coins into Bag A at a time. Student B must place 10 coins at a time into Bag B, and Student C must place 12 coins at a time into Bag C. What is the smallest number of coins all three bags can hold in common? (No “tricks” allowed. If a bag holds 16 coins, then we do not regard it as holding 5 coins; it’s holding 16 coins.)
Since 8 = 2·2·2, 10 = 2·5, and 12 = 2·2·3, the answer is their LCM, or 2·2·2·3·5, which is 120. But the best way to verify this is to use identical coins, and lay them down in three parallel rows laid, respectively, 8, 10, and 12 coins at a time. The students will see that the very first time all three rows “align” is at 120 coins.
Office Candy. The Candy-Bars-For-Offices club (or, CBfO) recently obtained 160 Hershey® Bars, 220 Kit-Kat® bars, and 300 Almond Joy® bars. What is the largest number of offices the CBfO can distribute all these items to, in such a way that each office will receive exactly the same number of each kind of candy bar, and how many of each kind would that be?
Since 160 = 2·2·2·2·2·5, 220 = 2·2·5·11, and 300 = 2·2·3·5·5, the answer is the GCF, or 2·2·5, which is 20. Each of the 20 offices would get 8 Hershey® Bars, 11 Kit-Kats®, and 15 Almond Joys®.
We’re likely not going to collect the actual candy, but we can model this by using, say, 160 dimes, 220 nickels, and 300 pennies (in real or play money). Interestingly, forming 20 “receiving groups” (each group getting 8 dimes, 11 nickels, and 15 pennies) does not demonstrate that 20 is the “largest” number we can use! However, a search for a greater number than 20 would soon reveal the significance of “common factors” since such numbers as 32 or 40 might divide 160, but won’t divide all three of 160, 220, and 300. Thus, the students should come to realize the importance of all three words in the phrase “greatest common factor” (or “greatest common divisor”). Moreover, it might lead to a discussion of the number of divisors possessed by any given composite. [For example, since 160 = (2^5)(5^1), it follows that 160 has only (5+1)(1+1) or 12 factors. (Why?) Those greater than 20 are 32, 40, 80, and 160. A quick check would verify that none of these values can act as the number of our “receiving groups” as none is a common factor of both 220 and 300 as well.]
Though more contrived than most MathLabs, each of these activities does indeed posit
Q. a practical question that
R. requires elementary mathematics to resolve, and
V. the results can be empirically verified for all to see.