Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 345 — #353
9.13. References 345
(e) Conclude that if f is an reverse-linear function and f .k/ D 0 for some k 2
Œ0; d/, then every positive integer is a divisor of f .n/ for infinitely many n.
(f) Conclude that every positive integer is a divisor of infinitely many Fibonacci
numbers.
Hint: Start the Fibonacci sequence with the values 0,1 instead of 1, 1.
Class Problems
Problem 9.34.
Find
remainder 9876 3456789 9 99 5555
6789 3414259 ; 14 : (9.27)
Problem 9.35.
The following properties of equivalence mod n follow directly from its definition
and simple properties of divisibility. See if you can prove them without looking up
the proofs in the text.
(a) If a b .mod n/, then ac bc .mod n/.
(b) If a b .mod n/ and b c .mod n/, then a c .mod n/.
(c) If a b .mod n/ and c d .mod n/, then ac bd .mod n/.
(d) rem.a; n/ a .mod n/.
Problem 9.36. (a) Why is a number written in decimal evenly divisible by 9 if and
only if the sum of its digits is a multiple of 9? Hint: 10 1 .mod 9/.
(b) Take a big number, such as 37273761261. Sum the digits, where every other
one is negated:
3 C . 7/ C 2 C . 7/ C 3 C . 7/ C 6 C . 1/ C 2 C . 6/ C 1 D 11
Explain why the original number is a multiple of 11 if and only if this sum is a
multiple of 11.
Problem 9.37.
At one time, the Guinness Book of World Records reported that the “greatest human