Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 347 — #355
9.13. References 347
More generally, we’ll say “k is a good modulus for base b” when, for any nonnegative
integer n, the sum of the digits of the base b representation of n is congruent
to n modulo k. So 2 is not a good modulus for base 10 because
763 6 7 C 6 C 3 .mod 2/:
(a) What integers k > 1 are good moduli for base 10?
(b) Show that if b 1 .mod k/, then k is good for base b.
(c) Prove conversely, that if k is good for some base b 2, then b 1 .mod k/.
Hint: The base b representation of b.
(d) Exactly which integers k > 1 are good moduli for base 106?
Problem 9.40.
We define the sequence of numbers
(
1; for n 3,
a n D
a n 1 C a n 2 C a n 3 C a n 4 ; for n > 3.
Use strong induction to prove that remainder.a n ; 3/ D 1 for all n 0.
Problems for Section 9.8
Exam Problems
Problem 9.41.
Definition. The set P of single variable integer polynomials can be defined recursively:
Base cases:
the identity function, Id Z .x/ WWD x is in P .
for any integer m the constant function, c m .x/ WWD m is in P .