What's the least positive integer,
n, having the following properties:
- n = (a^2)/2
- n = (b^3)/3
- n = (c^5)/5
(where a, b, and c are integers)
2n=a^2
3n=b^3
5n=c^5
So n must be a positive power of 2 multiplied by a positive power of 3 multiplied by a positive power of 5:
n = 2^x * 3^y * 5^z
where
x is odd
x is a multiple of 3
x is a multiple of 5
The smallest such x is 15
y is even
y is 1 less than a multiple of 3
y is a multiple of 5
The smallest such y is 20
z is even
z is a multiple of 3 and therefore of 6
z is 1 less than a multiple of 5
The smallest such z is 24
So n = 2^15 * 3^20 * 5^24
or n = 10^15 * 3^20 * 5^9
The 3^20 * 5^9 factor is 6810125783203125, so n is 6,810,125,783,203,125,000,000,000,000,000
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Posted by Charlie
on 2004-11-27 18:51:07 |
solution
