comute the following (a) 55555555 mod 5555 (b) 77766 mod 5 (c) Let n be a...
negate: (b) There exists a composite number n such p-11 (mod n) whenever p is a prime that doesn't divide n. (Recall that a natural number is called composite if it is not prime.) (c) For every integer n > 0, there exists a prime number p such that n S p < 2n. (b) There exists a composite number n such p-11 (mod n) whenever p is a prime that doesn't divide n. (Recall that a natural number is...
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.
ly(mod n). 2. Let n > 1 be an odd integer and suppose ? = y2 (mod n) for some x Prove that ged(x - yn) and ged(x + y, n) are nontrivial divisors of n.
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)
( i need Unique answer, don't copy and paste, please) Let N be an n-bit positive integer, and let a, b, c, and k be positive integers less than N. Assume that the multiplicative inverse (mod N) of a is a^(k-1) Give an O(n^3) algorithm for computing a^(b^c) mod N (i.e., a raised to the power b^c with the result taken mod N). Any solution that requires computing b^c is so inefficient that it will receive no credit.
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
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
2. Find 11644 mod 645 Use the following algorithm and show work! procedure modularExponentiation(b: integer, n = (ak-1ak-2...a1a0)2, m:positive integer) x:= 1 power := b mod m for i = 0 to k-1 If ai = 1 then x:= (x⋅power) mod m power := (power⋅power) mod m return x ( x equals bn mod m) Note: in this example m = 645, ai is the binary expansion of 644, b is 11.
Let n be a positive integer, and let s and t be integers. Then the following hold. I need the prove for (iii) Lemma 8.1 Let n be a positive integer, and let s and t be integers. Then the following hold. (i) We have s et mod n if and only if n dividest - s. (ii) We have pris + t) = Hn (s) +Mn(t) mod n. (iii) We have Hr(st) = Hn (3) Men(t) mod n. Proof....
Exercise 8 . Let n = 5-11-12 = 660. (A) Find i < y < n such that 1 mod(5), 3 mod(11), y 11 mod(12). (B) Suppose r E Z such that 4 mod(5) and 8 mod ( (11) Briefly explain why 55 divides +y, where y is the number from Part Show that, for all nEN, Exercise 9. 13 (29" -3")