Exercises 4.7 Exercises
1.
Give the sets of least absolute residues and least nonnegative residues for
2.
Prove that 13 divides
It is definitely worth while gaining intuition for modular manipulation by doing a bunch of examples.
3.
Compute
4.
Repeat Exercise 4.7.3, but with
5.
Repeat Exercise 4.7.3, but with
6.
Make up an exercise like Exercise 4.7.3 and dare a friend in class to solve it. (Make sure you can solve it before doing so!)
7.
Use the properties of congruence (in Proposition 4.3.2) or the definition to show that if
8.
Use the properties of congruence (in Proposition 4.3.2, not the definition) and induction to show that if
9.
Finish the details of proving Proposition 4.3.1, especially the second part (symmetric).
10.
Finish the details of proving Proposition 4.3.2.
11.
Find and prove what the possible last decimal digits are for a perfect square.
12.
Prove that if the sum of digits of a number is divisible by 3, then so is the number. (Hint: Write 225 as
13.
Prove that if the sum of digits of a number is divisible by 9, then so is the number.
14.
For which positive integers
15.
Complete the proof of Lemma 4.1.2 that having the same remainder when divided by
Consider Example 4.5.4 in these three extensions.
19.
Explore, using the interact after Question 4.6.7 or βby handβ, for exactly which moduli