Chapter 8 Number Theory220
Problem 8.11.
At one time, the Guinness Book of World Records reported that the “greatest human
calculator” was a guy who could compute 13th roots of 100-digit numbers that were
powers of 13. What a curious choice of tasks....
(a)Prove that
d^13 d .mod10/ (8.14)for 0 d < 10.
(b)Now prove that
n^13 n .mod10/ (8.15)for alln.
Problems for Section 8.6
Class Problems
Problem 8.12.
Two nonparallel lines in the real plane intersect at a point. Algebraically, this means
that the equations
yDm 1 xCb 1
yDm 2 xCb 2have a unique solution.x;y/, providedm 1 ¤m 2. This statement would be false if
we restrictedxandyto the integers, since the two lines could cross at a noninteger
point:
However, an analogous statement holds if we work over the integersmodulo a