# Answer to Question #23704 in Combinatorics | Number Theory for Lso

Question #23704
This question is with regards to Fermat Last Theorem and Number Theory. Let p and &theta; be primes with &theta;&gt;p such that p is not a factor of &theta;&minus;1. As &theta; is a prime, we know that for any x &isin; Z we can write x&theta;&minus;1 in the form x&theta;&minus;1 = k&theta;+1 for some integer k. Furthermore, since the greatest common divisor of p and &theta; &minus; 1 is 1, we can write 1 = ap + b(&theta; &minus; 1) for some integers a and b. (a) Show that every integer x can be written in the form x = yp + j&theta; for some integer y and some integer j. (b) Show that for p a prime, any prime &phi; that satisfies the conditions of Lemma 4.1 has the property that p is a factor of &phi; &minus; 1. (c) Prove the First Case of Fermat&rsquo;s Last Theorem for the exponents 13, 17 and 19. (That is, show that if there is a nonzero integer solution to xp + yp = zp for p = 13 then 13 is a factor of xyz, and so on.)
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