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MAS 4214/001 Elementary Number Theory, Fall 2014TEST 2Name:Date: 10/23/2013(1 point)Time allowed: 60 minutesShowALLsteps. Fifty points equal 100%.Assume in this test that all small letters represent integers andℓ≥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?