Math 328K (Daniel Allcock)
Homework 11, due Monday April 18, 2016
Problem 1. (Somtimes it’s hard to say what you mean) This problem is to
help remedy the disaster of 2/3 of the class not knowing what a prime number is on
exam II. Show that each of the following is NOT the definition of a prime number:
either exhibit a prime number that fails to satisfy the stated condition, or a nonprime number that does satisfy it. (A few of the “definitions” are ambiguous—just
point out the ambiguity since there is nothing else you can do with them.) In all
cases we assume p to be an integer.
(a) “p is prime if for all integers q < p with q 6= 1, q - p.”
(c) “p is prime if it is divisible by only itself and 1.” [According to this definition,
there is only one “prime” number! Say which it is.]
(d) “p is prime if it is positive and only divisible by one and itself.”
(e) “p is prime if it is larger than 1 and only divisible by one and itself.”
(f) “p is prime if it is larger than 2 and for every integer n < p, n 6= 1, n - p.”
(g) “p is prime if for all positive integers x, x < p, x - p.”
(i) “p is prime if p > 0 and p has only one positive factor other than 1, and that
is itself.” [Hint: hmmmm...is 1 a prime according to this “definition”?]
(j) “p is prime if p 6= 0, ±1 and p is only divisible by ±1 and ±p.”
(k) “p is prime if p > 1 and p is divisible by 1 and itself.”
Problem 2. Find the element x of Z8·7·5 satisfying the three congruences
x≡3
(mod 8)
x≡6
(mod 7)
x≡2
(mod 5).
Problem 3. Find all solutions to the polynomial congruence x2 ≡ 2 mod 119. (There
are four of them.)
Problem 4. Find all solutions to the polynomial congruence x3 − 4x2 − 2x − 83 ≡ 0
mod 1183. (Hint: you will need to use Hensel’s lemma on one of the smaller
congruences that you get by breaking apart this big one. Once you solve the
smaller congruences, you can crank on the Chinese remainder theorem in the usual
way.)
Problem 5. Say how many solutions there are to each of the following congruences.
Do not look for actual solutions. The point is that huge scary moduli are not
necessarily scary.
(a) x3 ≡ 101 modulo 3100 · 97100 · 101100 .
(b) x2 + x + 1 ≡ 0 modulo 7100 13100 .