Reading Quiz Section 3.1
1. Which of the following connectives makes the following true for any integers a, b and n > 0?
a ≡ b (mod n) a = b
(a) =⇒
(b) ⇐=
(c) ⇐⇒
(d) ∧
2. Let m, n ∈ Z where n > 0. Is it possible that there are multiple pairs of integers q and r such
that m = qn + r and 0 ≤ r < n?
(a) It is never possible.
(b) It is sometimes possible, depending on what m and n are.
(c) It is always possible.
3. Which of the following are true statements for all integers a, b and n > 0? Select all that apply.
(a) a is congruent to exactly one of 0, 1, . . . , n − 1 modulo n.
(b) a can be congruent to more than one of 0, 1, . . . , n − 1 modulo n.
(c) a is divisible by n if and only if a ≡ 0 (mod n).
(d) n ≡ 0 (mod n).
Practice Problems Section 3.1
1. Use the Division algorithm to show that any prime number p ≥ 5 must have remainder 1 or 5
on division by 6. Use this to show that p
2
+ 2 is composite for all such primes p.
Video Solution
2. Find the remainder of 57
33
+ 42
100
upon division by 6.
Video Solution
3. Prove that n
2
≡ 0 (mod 4) or n
2
≡ 1 (mod 4) for all n ∈ Z.
Video Solution
