Maths Item of the Month – October 2012
Factor in all the solutions
Factoring xn 1 (over the integers) for the first few positive integer values of n gives:
x1
x2
x3
x4
x5
1
1
1
1
1
x
(x
(x
(x
(x
1
1)( x 1)
1)( x 2 x 1)
1)( x 1)( x 2 1)
1)( x 4 x3 x 2 x 1)
Will the factors of xn 1 always have coefficients of
integer?
1 , 0 or 1 when n is a positive
Solution
The first counter-example exists at n 105 . Factorising x105 1 gives the factor:
( x 48
x 47
x 46
x 43 x 42 2 x 41 x 40
x 22
x 20
x17
x16
The coefficients of
x15
x14
x13
x39
x12
x36
x9
x35
x34
x8 2 x 7
x33
x6
x5
x32
x2
x31 x28
x26
x24
x 1)
2 are highlighted in red.
You may like to consider what special properties are possessed by the number 105, and
why they cause this to occur.
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JS 08/04/13
Final Version