number theory: homework help

Math 115A, SS2 2021
Due: Today Friday 09/10 at 18:00
This homework must be solved individually, and you are required to write out the honor
statement below by hand and sign it, and to upload it with your solutions.
HONOR STATEMENT: By signing this paper, I hereby declare that I solved this review
homework by my own, without any external collaboration (like friends, internet solutions, etc). If
needed, I am allowed to use our lecture notes only. Suspected misconduct on this exam will be
reported to the Office of Student Support and Judicial Affairs and, if established, will result in
disciplinary sanctions up through Dismissal from the University and a grade penalty up to a grade
of ‘F’ for the course.
To deliver: Submit your solutions on Gradescope. Each problem is worth 25 points, thus the
review homework is worth 100 points.
1. (a) Either, find x, y ∈ Z such that 557x + 325y = −2, or show that this equation has no
solutions x, y ∈ Z.
(b) Find all incongruent solutions to the congruence
18x ≡ 30 (mod 78).
2. (a) Let n ∈ N. Show that 10n − 1
is an integer.
(b) Let p 6= 2, 3, 5 be a prime number. Show that p divides 10p−1 − 1
(c) Show that every prime p 6= 2, 3, 5 divides a number of the form 11 · · · 1.
3. Let n ∈ N. Show that if n is not divisible by 3 or is divisible by 9, then
13 ≡ n (mod 45)
4. Solve the system of equations and find the least non-negative solution.

4x ≡ 1 (mod 5)
3x ≡ 2 (mod 11)
2x ≡ 3 (mod 13)
Write your answer both as a congruence class and in its parametric form.

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