Math 110 Problem Set 2 Solutions

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Math 110 Problem Set 2 3 be prime. Show that the only Solutions tox2 1 (modp) arex 1 (modp).Solution:Note that we have(x+ 1) (x 1) x2 1 0 (modp)Thus, by Exercise 7(a), we have that eitherx 1 0 (modp) orx+ 1 0(modp) so we havex 1 (modp) as 2 (mod 7) andx 3 (mod 10). What isxcongruent to mod70?Solution:This is easiest to do by trial and error. The 2nd condition meansthatxis 3, 13, 23, 33, 43, 53, or 63 modulo 70. Checking, we see that the onlyone of these which is 2 modulo 7 is 23, soxmust be 23 (mod 70). Note thatthe Chinese Remainder Theorem guarantees that there is a unique answer tothe prime. Show thatap a(modp) for :Let us consider two cases:pdividesa, andpdoes not dividea. Ifpdividesa, then both sides are 0 modulop.

Math 110 Problem Set 2 Solutions 3.8 Let p ‚ 3 be prime. Show that the only solutions to x2 ...

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