Find the remainder when r^13 + 1 is divided by r-1.
Performing synthetic division, we have
1 [ 1 0 0 0 0 0 0 0 0 0 0 0 0 1 ]
1 1 1 1 1 1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1 1 1 1 1 2
The remainder is 2
Geometric sequence:
1+r+r2+r3+r4+…+r12=r13−1r−11+r+r2+r3+r4+…+r12=r13r−1−1r−1|+2r−11+r+r2+r3+r4+…+r12+2r−1=r13r−1−1r−1+2r−11+r+r2+r3+r4+…+r12+2r−1=r13r−1+1r−11+r+r2+r3+r4+…+r12+2r−1=r13+1r−1
The remainder is 2.