As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

## Saturday, November 26, 2011

### M381 'Challenge Exercise'

Each Open University M381 Number Theory booklet has a number of 'challenge exercises'. This is one of them.

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.

1. Hi Nilo,

I 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.

1. Paddy, I'm pretty sure you're correct and that yours is the smallest solution.

Working:
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? :)

2. 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 :)

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)