MTH 182
(1 pt) Section: 10 AM or 11 AM or 1 PM Name _______________________
Worksheet 04: The Division Algorithm, Primes and Congruences V1
1. (6 pt) For each of the following pairs of integers determine the values ! m div n and ! m mod n .
a) m = 1000 , n = 205
b) m = −1000 , n = 205
c) m = 1000, n = 2000
2. (8 pt) Let ! p > 3 be a prime integer.
(a) Prove that either ! p mod 6 = 1 or ! p mod 6 = 5 .
(b) Prove that ! p 2 ≡ 1( mod 6 ) .
MTH 182 Worksheet 04: The Division Algorithm, Primes and Congruences V1
3. (10 pt) Assume ! n ∈! . Using the proof the Theorem 7.22 as a guide, show ! 5 n 2 if and only if ! 5 n .
Extra Credit (8 pt). Let n be a positive integer. Prove that if the alternating sum of its digits is divisible
by 11, then ! 11 n For example, if ! n = 924 , then ! 9 − 2 + 4 = 11 = 1⋅11 and ! 924 = 11⋅ 84 . Also, if
! n = 123321 , then ! 1− 2 + 3 − 3 + 2 − 1 = 0 = 0 ⋅11 and ! 123321 = 11⋅11211 .