I got this in my calc class although I don't think its calculus and I didn't know where to post this. How do you find the largest product possible if factors add up to a given sum. For example, 27. I remember doing this sort of thing years ago but can't find it online or in books or a search on this site.
My interpretation of your problem statement is to find the largest product of the factors of a number that sum to a given number. Clearly, the largest product of the factors of a number would be the product of all the factors.
The real problem would appear to be finding the greatest product of a series of numbers all of which have the same sum.
For instance, the factors of 16 and 25 sum to 31. The product of 1, 2,4, 8 and 16 is 1024 while the product of 1, 5 ad 25 is 125.
Similarly, the factors of 20 and 37 sum to 38. The product of 1, 2, 5, 10 and 20 is 2000 while the product of 1 and 37 is 37.
Consider the numbers 24 and 38 whose factors sum ro 60. The product of 1, 2, 3, 4, 6, 8, 12 and 24 is 331,776 while the product of 1, 2, 19 and 38 is 1444.
Consider the numbers 30 and 45 whose factors sum to 72. The product of 1, 2, 3, 5, 6, 19, 15 and 30 is 810,000 while the product of 1, 2, 23 and 46 is 2116.
It would appear that given one finds a series of numbers whose factors sum to a specified constant, the one that produces the largest product would be the one with the greatest number of factors.
I do not know if this would hold up under a more exhaustive analysis.
Any constructive comments are welcomed...
