25. Show that the number of reduced residues a (mod m) such that am-1 1 (mod...
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0
8. Given integers m and 1<a<m, with am, prove that the equation ar = 1 (mod m) has no solution. (This means that here is no 1 appearing in the multiplication table mod m, in front of any of the divisors of m. That is, if m is composite, and a is a factor of m then a has no multiplicative inverse in mod m.)
please answer question #5 and show steps
5. Solving Quadratics (mod p). Use #1 (a) above and the quadratic formula (mod p) to find a pair of solutions (if possible) for each of the following quadratic equations (1), 2r2 +3 -4- (mod 7) (ii), 3r2-2r +1 0 (mod 19) (ii). 3z2 2r -0 (mod 23) 1. Euler's Criterion. (a). Use Euler's Criterion to the Legendre symbols below: (iv). (10/23) (b). Assume a is a quadratic residue mod p, and assume...
Let m be a positive integer. Show that a mod m - b mod m t a - b (mod m) Drag the necessary statements and drop them into the appropriate blank to build your proof (mod m Dag the mecesary eemnes a ohem int the approprite Proof method: Proof assumptions), at-qm + Proof by contradiction aaandh mam it Implication(s) and deduction(s) resulting from the assumption(s): a mk + bmk Hqm tr a-(k + q)m+ r Conclusion(s) from implications and...
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer, and let m = 6q+r where q and r are integers with 0 <r < 6. Use (a) and rules of exponents to show that 2" = 2 mod 9 (c) Use (b) to find an s in {0,1,...,8} with 21024 = s mod 9.
1. Show that a 1728 = 1 (mod p) when p= 7, 13, 19 for all a E N such that p /a. 2. Let p be a prime and p = 3 (mod 4). Show that r2 = -1 (mod p) has no solution. (Hint: Raise both side to (p-1)/2.)
We know that we can reduce the base of an exponent modulo m: a(a mod m)k (mod m). But the same is not true of the exponent itself! That is, we cannot write aa mod m (mod m). This is easily seen to be false in general. Consider, for instance, that 210 mod 3 1 but 210 mod 3 mod 3 21 mod 3-2. The correct law for the exponent is more subtle. We will prove it in steps (a)...
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai?
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
Find (i) 2^25 mod 21, (ii) 7^66 mod 120 and (iii) the last two digits of 1 + 7^162 + 5^121 * 3^312
mod use Show that the following equations have no integer solutions: (4) 25 + 12x2 + 24x + 1 = 0.