The GCF and the LCM of two prime numbers, p and q are 1 and 91 respectively. Find the value of p+q.
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Answer:
To find the least common multiple, LCM, of any two (or more) numbers, we simply factor each of them into the product of primes. Then, the LCM of the two (or more) numbers is the product of as many of those prime factors as are needed to reproduce each of the original numbers.
For example, to find the LCM of 18 and 24, we start by factoring each of them into a product of primes. Thus, 18 = 2*3*3, and 24 = 2*2*2*3. Between the two factorizations, we have a total of four 2’s and three 3’s. However, to reproduce either of 18 and 24, we need only three 2’s and two 3’s. Therefore the LCM of 18 and 24 is the product of those three 2’s and two 3’s. LCM(18,24) = 2*2*2*3*3 = 72. As to the single 2 and 3 we didn’t need for the LCM (sort of like spare parts in a manufacturing process), their product, 6, is the greatest common divisor (GCD) of 18 and 24.
Note that the product of all four of the 2’s and the three 3’s is simply the product of 18 and 24, However, we have also just seen that that same product is also the product of the LCM and the GCD of 18 and 24, our original numbers. That is 18*24 = 2^4*3^2 = 432 = 72*6. From the way we uncovered this fact, it should be obvious that this was no coincidence. The product of the LCM and the GCD of any numbers is always equal to the product of the numbers themselves.
After that long aside, let me get back to your question. Instead of any two ordinary, garden variety numbers like 18 and 24, you want the LCM of two prime numbers. Since they are prime, they don’t submit to factoring, so their LCM is simply their product. Also, since they are prime, their GCD is 1, so the result we arrived at earlier still holds; m*n = LCM(m,n)*GCD(m,n), for any positive integers m and n.