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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.

      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 :)


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