Two members have a GCF of two times 11 and an LCM of 13 squared times two times 11. If one of those numbers is 13 squared times two times 11, what is the other number?
First of all, let’s think about what we know about the GCF and the LCM. The greatest common factor, the GCF, of two whole numbers is the largest whole number which is a factor of both. The lowest common multiple, LCM, of two whole numbers is the smallest whole number which is a multiple of both. We’ve been given one of those numbers, the GCF and the LCM, all in factored form. Our first number is 13 times 13 times two times 11, and the GCF is two times 11. Since the GCF is the largest whole number which is a factor of both, when we list out the factors, the GCF will be all of the values in both lists. That means if two times 11 is the GCF, then both two and 11 will be factors of our second number.
Now, the LCM is a bit different. Since it’s the smallest whole number which is a multiple of both, the LCM are all of the factors found in either list. This means that the LCM lists all the possible factors for our second number. Our second number cannot have any factors besides 13, two, and 11. And because of the GCF, we know that our second number has factors of two and 11. Now we only have to decide, does our second number include a factor of 13 or not? If we thought, well, maybe the second number has one factor of 13, well, this would not work because if we add 13 as a factor of our second number, then the GCF would have to be 13 times two times 11. Since the only shared factors are two and 11 and all of the factors are 13, two, and 11, our second number must be two times 11. It must be 22.