Simple? math question

Well, last night I got stumped by my 6th grader’s homework. Wondering if any of you less math-challenged folk out there could help me out.

Apparently they are studying factoring, and this particular assignment concerned greatest common factors (GCF) and lowest common multiples (LCM). In one section you were to find the pair of numbers that shared the specified attributes. The first question specified LCM = 35, GCF = 7.

Am I wrong in thinking that those were the pair of numbers requested? The LCM of 7 and 35 = 35, and the GCF of 7 and 35 = 7.

Seemed like the same was true for the other 2 sets of criteria. I remember one was GCF = 10, LCM = 120. Can’t remember the other set. I remember the GCF was 1, and think the LCM may have been 60.

Seemed kinda - um - stupid. Dammit, Jim! I’m a LAWYER, not a mathemetician!

I could only guess that this was an attempt to present these concepts in a different manner, which might enlighten those kids that did not grasp the more common presentations.

Any thoughts?

If the LCM(x,y) = 35 and the GCF(x,y) = 7, then x and y must both be multiples of 7 and both must also be factors of 35. The only factors of 35 are 1, 5, 7, and 35. Only 7 and 35 are multiples of 7, so those two numbers are x and y.

If the LCM(x,y) = 120 and the GCF(x,y) = 10, then x and y must both be multiples of 10 and both must also be factors of 120. So both must be in the set {10, 20, 30, 40, 60, 120}. If you try all the pairs in that set you’ll see that x and y can be 10 and 120 or it can be 30 and 40. All the other pairs have a different LCM.

If the LCM(x,y) = 60 and the GCF(x,y) = 1, then x and y must both be factors of 60. They must be in the set {1,2,3,4,5,10,12,15,20,30,60}. If you try all the pairs, you’ll see that x and y can be 1 and 60 or they can be 3 and 20 or they can be 4 and 15 or they can be 5 and 12.


The other thing that’s helpful for checking this type of homework:

GCF(x,y) * LCM(x,y) = x * y

That is, if you multiply the GCF and LCM, it’s the same product as when you multiply the two numbers. (Not useful to the question at hand, but kind of interesting to the kids, usually.)

Wendell answered the question about the numbers really well, and there’s no reason to repeat that, but to address your other question, about why those numbers were chosen:

First, keep in mind that when making up the assignments, the teacher may have just chosen x and y, and only later realized that they are then equal to the LCM/GCF. On the other hand, I can see a good use for it: sometimes, you want to show, in the homework, the “limiting conditions” of a concept. In this case, when the GCF and LCM of two numbers are those two numbers. (And I guarantee you, if one of the numbers is the GCF of the 2, the other number will be the LCM, and vice versa.)

And if the other problem was either {10,120} or {1,60}, as Wendell noted, the answers aren’t necessarily the original answers, so it’s not as if the teacher was always following the pattern of having that be the ONLY answer. As for that being ONE of the answers, I want to point out this:

  • For any 2 numbers (x and y), the GCF(x,y) [call this g] will ALWAYS be a factor of the LCM(x,y) [call this l], and
  • Given those 2 numbers (g, l), LCM(g, l) = l, and GCF(g, l) = g, so
  • Given any GCF/LCM set, one of the solutions for the original number will always be g and l. (It will sometimes be the only solution.)

So when given a problem like you describe, it isn’t surprising that you came up with the original numbers as a possible solution.

In my ignorance, (and I think as the question was phrased) I suspected there was only one possible pair as the solution to each question. So when I realized it worked for the original numbers, my thought was “Now that was a stupid assignment!” And with a lot of these types of questions, I find I am able to figure them out through a sort of intuitive trial and error, but I don’t know of the specific formulas/approaches to use.

It is always a boost to the old ego, when your sweet child asks, “Daddy, will you help me with my homework?” and I find myself answering, "Well, I WOULD if I COULD!"

Now, about that B we - I mean my eldest daughter - got on that Odysseus paper…

It’s actually better to work them out by “intuitive trial and error,” since that means that you’ll understand the underlying concepts better. Afterwards, you can then learn the formulas that explain how to do faster. If you understand the idea but not the formula, you can work out the formula. If you know the formula but don’t understand where it came from, you’re in trouble if you forget the formula. Understanding is always better than memorization.