Half of the number we are looking for is a square, a third of the number is a cubic and lastly a fifth is a fifth power. Find a number satisfying these properties.
Solution on request.
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Half of the number we are looking for is a square, a third of the number is a cubic and lastly a fifth is a fifth power. Find a number satisfying these properties.
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Hi Nilo,
ReplyDeleteI informally played with the numbers to get a solution of:
2^(15) 3^(10) 5^(6)
I'd be interested to see a full solution.
Paddy, I'm pretty sure you're correct and that yours is the smallest solution.
DeleteWorking:
n must be divisible by 2,3,5
Therefore n = 5^a.3^b.2^c
(2^c*3^b*5^a)^(1/n) = 2^(c/n)*3^(b/n)*2^(a/n), so the powers of 2,3,5 can be done separately.
Where m,n,o are integers (not necessarily consistent for a,b,c)
a-1,b,c = 5m
b-1,a,c = 3n
c-1,a,b = 2o
a = 5m+1, 3n, 2o
a is divisible by 6
5*1+1 = 6, so 6 is the smallest value for a
b = 5m, 3n+1, 2o
b is divisible by 10
3*3+1 = 10, so 10 is the smallest value for b
c = 5m, 3n, 2o+1
c is divisible by 15
2*7+1 = 15, so 15 the smallest value for c
Using these: n = 2^(15) 3^(10) 5^(6)
For a general solution: x = n*z^15 (where n is the solution shown and z is any positive integer)
...does 0 count? :)
Hey Nilo, I think i have a general solution and I want to see if it lines up with what you have, can I please have what you got? Thanks :)
ReplyDelete