MATH 255: ELEMENTARY NUMBER THEORY
HOMEWORK #5
JOHN VOIGHT
4.3: The Chinese Remainder Theorem
Problem 4.3.4(c). Find all the solutions to the following system of linear congruences:
x≡0 (mod 2)
x≡0 (mod 3)
x≡1 (mod 5)
x≡6 (mod 7).
Problem 4.3.12. Solve the following ancient Indian problem: If eggs are removed from a
basket 2,3,4,5,6 at a time, there remain, respectively, 1,2,3,4,5 eggs. But if the eggs are
removed 7 at a time, no eggs remain. What is the least number of eggs that could have been
in the basket?
Problem 4.3.14∗. Show that if a, b, c ∈Zhave gcd(a, b) = 1, then there is an integer n
such that gcd(an +b, c) = 1.
Problem 4.3.A. Let f(x)∈Z[x] be a polynomial with integral coefficients. For m∈Z>1,
let #X(Z/mZ) denote the number of solutions in Z/mZof the congruence
f(x)≡0 (mod m).
(a) Prove that if m=m1m2, where gcd(m1, m2) = 1, then
#X(Z/mZ) = #X(Z/m1Z)·#X(Z/m2Z).
(b) What can you conclude if gcd(m1, m2)>1?
4.4: Solving Polynomial Congruences
Problem 4.4.1. Find all the solutions of each of the following congruences:
(a) x2+ 4x+ 2 ≡0 (mod 7)
(b) x2+ 4x+ 2 ≡0 (mod 49)
(c) x2+ 4x+ 2 ≡0 (mod 343)
Problem 4.4.10. How many incongruent solutions are there to the congruence x5+x−6≡0
(mod 144)?
Problem 4.4.A. Let k∈Z>0.
(a) Show that the product of any kconsecutive integers is divisible by k!. [Hint: Use a
binomial coefficient.]
Date: Due Wednesday, 25 February 2009.
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