Prove the following: a. a = b (mod b) implies b =a (mod n) a = b (mod n) and b = c (mod n) imply a = c(mod n)
10.) Consider (Zn; n), the set Zn with mod n multiplication. i. Argue that if neither a nor b has any common divisors greater than 1 with n then neither does ab. [Equivalently gcd(a; n) = 1, etc.] ii. Argue that if a does not have any common divisors greater than 1 with n, then [a]n has a multiplicative inverse in Zn. iii. Argue that (i) and (ii) imply that the set of elements f[a]n 2 Znjgcd(a; n) = 1g...
Suppose a c mod n and bd mod n. (a) show that a + b c + d mod n (b) show that a * b c * d mod n. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
(b) Prove that n is an odd pseudoprime number if and only if 2"-1-1 mod n. (b) Prove that n is an odd pseudoprime number if and only if 2"-1-1 mod n.
I have first part of question good. Need to prove unique modulo and do not know where to start. Prove that the congruences x-a mod n and x b mod m admit a simultaneous solution if and only if (n, m) | (a -b). Moreover, if a solution exists, then the solution is unique modulo [m, n). Prove that the congruences x-a mod n and x b mod m admit a simultaneous solution if and only if (n, m) |...
Recall that: ged: NN → N gcd(a,0) = a. gcd(a,b) = gcd(b, mod(a,b)), if b > 0. and mod : Nx (N – {0}) ► N mod(a,b) = a if a <b. mod(a,b) = mod(a - b,b), if a > b. and fib: N → N fib(0) = 0 fib(1) = 1 fib(n) | if n >1=fib(n − 1) +fib(n - 2) Prove the following by induction. you cannot use any extra lemmas or existing results. Vn e N, ged(fib(n...
9. Suppose n : (Z4 ⓇZ3) + (Z2 Z3) is defined by n(a,b) = (a mod 2, 0). This is a homomorphism. (You can believe me about that, and you don't need to check.) Write down the elements of the kernel of n.
What is the smallest positive integer n that has the following characteristics? n mod 3=2,n mod 5=3, and n mod 7=5
comute the following (a) 55555555 mod 5555 (b) 77766 mod 5 (c) Let n be a composite integer bigger than 4, compute n-1 i mod n i=1
4. Show that the following congruence is true ab = (a mod n)" (mod n) for any positive integers a, b, and n.