Tag Archives: congruence

Homework Chp 5 Update – problem #20

Hi everyone,

Problem #20 in Chapter 5 uses an idea that we have *not* yet covered in class. Ā It is no longer a required problem – you don’t have to do it – but I will give you extra credit if you turn in a solution. Ā This is excellent practice in reading and applying a definition (just as we have been doing for the definitions of odd, even, divides, and so on). Ā The problem reliesĀ on a new definition, that of congruence mod n – itĀ appears in the book as Definition 5.1 on page 105, but I will also give it here:

Definition. Ā Given integers a and b, and n \in \mathbb{N}, we say “a is congruent to b mod n”, orĀ a \equiv b \pmod n, if
n | (b-a).

For example, if are told that x \equiv 7 \pmod 3, then we can conclude:
3|(7-x) Ā (by the definition of congruence), and
7-x = 3k for some integer k (by the definition of divides)

Hope this helps! Ā Please write back and let me know if you have any questions.

Best of luck,
Prof. Reitz