MAS 4214/001 Elementary Number Theory, Fall 2014
TEST 2
Name:
Date: 10/23/2013
(1 point)
Time allowed: 60 minutes
Show
ALL
steps. Fifty points equal 100%.
Assume in this test that all small letters represent integers and
ℓ
≥
2.
Question 1.(6 points)True or false: Explanation is not needed.(a) An integerfis a common divisor ofa,bifffis a divisor of gcd(a,b).(b) An integermis a common multiple ofa,biffmis a multiple of lcm(a,b).(c) We have the following equalities: gcd(42,70,63) = gcd(gcd(42,70),63) = gcd(14,63) = 7.(d) The number 824+ 1 is prime.(e) The integer 2369-1 is not prime because 369 is not prime.(f) Ifpis prime, then 2p-1 is prime.
Answer:
Question 2.(2+6 points) Consider the equation 51x+ 57y=c. It is known that gcd(51,57) = 3 and that(51)(9) + (57)(-8) = 3.(a) For which values ofcdoes the equation have an integer solution?