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I am making a largish number fraction Puzzle and need help with numerators?
Background: I was doing some LCD examples and contrived 1/27, 1/37, 1/77, and 1/117 to show that one can't just list multiples. (for 12 and 15 we can list 12,24,,36,48,60 and 15,30,45,60 and easily get the LCD).
I want to solve i/27 + j/37 + k/77 + l/177 = 1. This is what I believe is called a Diophantine Equation (integer solutions and may have more than one solution).
Excel doesn't want to find an answer for me and since I made the puzzle google isn't working for me. I want i,j,k, and l to be reasonable numbers. Best answer to the most reasonable numbers.
since there are 7s in the denominators it seems there should be a bonus for 7s in the numerators.
I intend to put this in a puzzle magazine and will cite you using your yahoo id or any name (I can say to my mother.) you want me to use.
Thank you DW. It should be 117. Then the LCD is 999,999.
To wit, solve for integers i,j,k,l i/27 + j/37 + k/77 + l/117 = 1.
(It also occured to me that 2, or a half, or a third would also work.
I am running a program. It gave the answer:
3/27 + 0/37 + 77/77 - 13/117 = 1.
Using 77/77 and 0/37 are so boring I can't use them in a puzzle. And notice that 3/27 = -13/117, so that is boring too.
1 Answer
- DWReadLv 77 years ago
Are you using 117 or 177? You list both.
Factor each denominator into powers of primes
27 = 3³
37 = 37
77 = 7×11
177 = 3×59
Least common denominator = 3³×37×7×11×59 = 4,538,457
