It is very easy to find a solution. 7 = 3+3+1; so 3/15, 3/15, 1/15 becomes 1/5 + 1/5 + 1/15. So [5, 5, 15] is a solution.
But I resorted to a program to find (hopefully) all the solutions.
{x,y,z} can be in any order, but for brevity, will show only x <= y <= z.
LCM [x, y, z]
15 [3, 15, 15]
15 [5, 5, 15]
30 [3, 10, 30]
30 [5, 6, 10]
45 [3, 9, 45]
60 [4, 5, 60]
60 [3, 12, 20]
60 [4, 6, 20]
120 [3, 8, 120]
----------
maxmultiplier = 10000
solutions = []
for mul in range(1, maxmultiplier):
mylcm = 15*mul
mysum = 7*mul
myfacs = factors(mylcm)
for comb in combinations_with_replacement(myfacs,2):
a = comb[0]
b = comb[1]
c = mysum - a - b
if c not in myfacs:
continue
x = round(mylcm/a)
y = round(mylcm/b)
z = round(mylcm/c)
ans = sorted([x,y,z])
if ans not in solutions:
solutions.append(ans)
print(mylcm, ans)
|
|
Posted by Larry
on 2024-08-10 14:56:45 |
Solution
