>>14252694Express each number as its prime factorization. Throw out 1 as the trivial factor, the empty product of no primes (as they say), neither prime nor composite. There are fifteen elements to inspect, of which six (2, 3, 5, 7, 11, 13) are prime. Of the remaining nine, note that 9 = 3^2 and 16 = 2^4. Then notice that the other seven elements are either lower powers of two, products of two primes already expressed, or in the case of twelve, a product of "three" primes already entailed in the above: 2 x 2 x 3. In other words, they can be tossed as redundant.
We wish to find LCM(1-16). It is equal to the smallest prime factorization which can be "assembled" from the above, which is
LCM(1-16) = 2^4 * 3^2 * 5 * 7 * 11 * 13 = 144 * 35 * 143 = (4320 + 720) * 143 = 5040 * 143 = 720720,
Where the above was worked with a blend of mental math and pen-and-paper. It turns out that 720720 is a superior highly composite number, and my arbitrary choice of multiplying above also led to a 5040 term, itself a notable factorial number.
The term 720720 suggests 720720 = 6! * 1001 = 6! * (7 * 11 * 13) = (2 * 3 * 2^2 * 5 * 2 * 3)(7 * 11 * 13), a rephrase of the above.
