By Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin
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Extra info for Algebra and number theory : an integrated approach
Definition. Let A= [aij] and B = [b;j] be matrices in the set MkxnOR). The sum A+ B of these matrices is the matrix C = [c;j] E MkxnOR), whose entries are C;j = a;j + b;j for every pair of indices (i, }), where 1 ::; i ::; k, 1 ::; j ::; n. The definition means that we can only add matrices if they have the same dimension and in order to add two matrices of the same dimension we just add the corresponding entries of the two matrices. In this way matrix addition is reduced to the addition of the corresponding entries.
We continue this process; thus if rj-! and rj have been obtained with rj =f. , rj+! such that rj-! = rjqj+! + rj+! with 0 _:::: rH 1 < rj. Since, at each step, rj-! < rj, then this process will terminate in a finite number of steps so that at some step rk = 0. We obtain the following chain of equalities: + r,, b = r1q2 + r2, r, = r2q3 + r3, ... , rk-3 = rk-2qk-! qk + rk. rk-! 4) It is now possible to prove, using the Principle of Mathematical Induction, that rk is a common divisor of a and b. 4 we finally see that rk divides a and b.
15. Suppose that 2n + 1 is a prime where n is a positive integer. Prove that n = 2k for some positive integer k. 16. Let n, k be positive integers. Let n = kq that GCD(n, k) = GCD(k, k- r). +r where 0:::; r < k. 17. Find a positive integer k such that 1 + 2 + · · · + k is a three-digit number with all digits equal. 18. The coefficient of x in the third member of the decomposition of the binomial (1 + 2x)n is 264. Find the member of this decomposition with the largest coefficient. 19. ) 2 ::=: kk for all positive integer k.
Algebra and number theory : an integrated approach by Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin