Exam 3 Solutions on Elementary Number Theory - MAS...

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MAS 4214/001 Elementary Number Theory, Fall 2014TEST 3Name:Date: 12/01/2014(2 points)Time allowed: 60 minutesShowALLsteps. Fifty points equal 100%.Assume in this test that all small Roman letters represent integers, andm1.Question 1.(5 points)True or false: Explanation is not needed. All small case letters standfor integers.(a) An integern2 is prime iff (n1)!≡ −1 (modn).(b) Assume thata, bare positive integers. Ifas+bt= 1, thensis an inverse ofamodulob, andtis aninverse ofbmoduloa.(c) Suppose thatm, nare positive integers wihm|n. Thenab(modm) implies thatab(modn),but the converse does not hold always.(d)φ(43·57) = (4342)(5756).(e) Suppose thatm2. If 7m-1negationslash≡1 (modm), thenmis composite.Answer:(a)T(b)T(c)F(d)T(e)T
Question 2.(3+3 points)It is given that 1031 (mod 37).(a) Prove that ifa= 109x+ 106y+ 103z+w, thenax+y+z+w(mod 37).

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