## Multiplicative median
However, it breaks down when you have three numbers. As an example, I will use 20, 24, and 32. The product is 15360. The GCD is 4. The LCM is 480. This time, GCD * LCM = 1920. In fact, if abc = LCM(a,b,c) * GCD(a,b,c), it can be proven that a and b are coprime, a and c are coprime, and b and c are coprime. Furthermore, in this case the GCD = 1 and the LCM = abc. The proof of that theorem requires a new function discovered by me called the multiplicative median. MM(a,b,c) = abc / (GCD(a,b,c) * LCM(a,b,c)). For example, MM(20,24,32) = 15360 / 1920 = 8. By definition, GCD(a,b,c) * LCM(a,b,c) * MM(a,b,c) = abc. It can also be proved that GCD(a,b,c) <= MM(a,b,c) <= LCM(a,b,c). This lemma will be proved below. Now, the above theorem can be proved as follows: abc = LCM(a,b,c) * GCD(a,b,c) = LCM(a,b,c) * GCD(a,b,c) * MM(a,b,c). Thus MM(a,b,c) = 1. Now, 1 divides every positive integer so GCD(a,b,c) >=1 Also, GCD(a,b,c) <= MM(a,b,c) = 1. Thus GCD(a,b,c) = 1. Also, abc = GCD(a,b,c) * LCM(a,b,c) = LCM(a,b,c). Thus LCM(a,b,c) = abc. Now, let a and b have a common factor greater than 1. Then GCD(a,b) > 1 so LCM(a,b) = ab / GCDa,b < ab. Now, LCM(a,b,c) = LCM(LCM(a,b),c) <= c * LCM(a,b) < c * (ab) = abc. Thus LCM(a,b,c) < abc. However, LCM(a,b,c) = abc so we have a contradiction. Thus a and b are coprime. Similarly, a and c are coprime and so are b and c. Thus the theorem is proved. I have discovered many more things about this curious MM function. I
have thought about when abc = MM(a,b,c) The properties come from the prime factorisation of a, b, and c. For each prime, a has a certain power of that prime, as do b and c. Now, LCM(a,b,c) has the maximum power of those three powers, GCD(a,b,c) has the minimum, and abc has the sum, so MM(a,b,c) has the median of the powers. I chose the name "multiplicative median" because it has the median prime factorisation. As a consequence of this, GCD(a,b,c) <= MM(a,b,c) <= LCM(a,b,c), which proves the lemma from earlier. If you really want the list of 27 values of c, here it is: 9; 72; 243;
576; 1125; 1944; 6561; 9000; 15,552; 30,375; 52,488; 72,000; 140,625;
243,000; 419,904; 820,125; 1,125,000; 1,944,000; 3,796,875; 6,561,000;
9,000,000; 30,375,000; 52,488,000; 102,515,625; 243,000,000; 820,125,000;
6,561,000,000. If you really, REALLY love crunching numbers and have nothing
else to do, try checking them all to see that if a = 810,000 and b = 72,900,
MM(a,b,c) ## Related entries•Coming soon | ||

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