MTH 200 – Graded Assignment 4
Material from Module 7 (Divisibility) and Module 8 (Prime Numbers). The bulk of this is
coming from Module 7; Module 8 involves proofs of several results that we'll pick up and
apply a bit later on, but not that much to do in the way of problems.
P1: True or false?
a) 15 | ( 5)
b) 17 | 21
c) If a | b , then a is a multiple of b .
P2: Let a , b , and c be integers. Prove that if a | b and a | c , then a | (mb  nc) for any
integers m and n .
P3: Let a , b , c , and d be integers. Prove that if a | b and c | d , then ac | bd .
P4: For each pair of integers a and b given below, find q and r as in the Division
Algorithm, and express a as a  qb  r .
a) a  171 , b  22
b) a  354 , b  61
P5: Let a , b , q , and r be as in the Division Algorithm: a  qb  r , with b  0 and
0  r  b . Suppose that a  0 . Prove that
a  (q  1)b  (b  r )
P6: Compute:
a) 85 div 12
b) 85 mod 12
P7: Use the method of prime factorization to find the greatest common divisor and least
common multiple of the numbers 7448 , 2166 , 2051
P8: For a  1519 and b  1240 , use Euclid’s algorithm to find gcd(a, b) . Then, use the
result that relates gcd(a, b) and lcm(a, b) to calculate lcm(a, b) .
P9: Let a , b , and c be integers. Suppose that a | b and c | b . Suppose also that
gcd(a, c)  1 . Prove that ac | b . (Give some examples before proving the result.)
Proof outline:
 Suppose that a | b . The definition of divides immediately gives you what?
 Now, suppose c | b and don’t write a definition of divides for that one. Instead,
apply it to the previous step: since c | b , c must divide whatever expression you
have for b above.
 Then, use that fact, the fact that gcd(a, c)  1 , and Euclid’s lemma on p. 18. What
must c divide?
 Now apply the definition of divides to that statement. Make a substitution,
regroup, and you’ll have it.
P10: Let a  47 , b  19 . Use Euclid’s Algorithm (the extended version) to express
gcd(47,19) in the form ma  nb  gcd(47,19) .
P11: Solve the Diophantine equation 47 x  19 y  4 . Give the general form of all integer
solutions.
P12: Explain why the Diophantine equation 189 x  45 y  16 does not have a solution.
P13: Prove that any prime of the form 3m  1 for some m¥ is also of the form
6n  1for some n¥ .
Hints/outline for this one appear in the section 1.4 suggested textbook exercises.
P14: Compute
a)  (373)
b)  (1841)
c)  (103823)
d)  (141032)
You can use the prime factorization applet linked in module 8 to quickly factor these.
Since that also serves as a phi calculator, please break your phi calculations down step by
step and justify each step by citing which formula/theorem you’re using.