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Chapter 1: Q12E (page 48)

What is 222006(mod3)?

Short Answer

Expert verified

The solution is 222006(mod3)=1.

Step by step solution

01

Introduction

A function that returns a number or variable's absolute value is known as a modulus function. The magnitude of the number of variables is produced. Another name for it is an absolute value function. No matter what input was provided to this function, a positive result is always the outcome.

02

Calculating 222006(mod3).

The value starts from n=0. i.e.,

For鈥�n=1,鈥夆赌�21=2(mod3)For鈥�n=2,鈥夆赌�22=1(mod3)For鈥�n=3,鈥夆赌�23=2(mod3)For鈥�n=4,鈥夆赌�24=1(mod3)

The even value ofn remainder is1 and for odd it is 2. i.e.,

22n=1(mod3)22n+1=2(mod3)

Therefore, the solution is222006(mod3)=1 .

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Most popular questions from this chapter

Show that log(n!)=胃(nlogn)

(Hint: To show an upper bound, compare (n!) with nn. To show a lower bound, compare it with (n2)n2).

Calculate 2125mod127using any method you choose. (Hint: 127 is prime.)

Give a polynomial-time algorithm for computing,abcmodp given a,b,c, and prime p.

The grade-school algorithm for multiplying two n-bit binary numbers x and y consist of addingtogethern copies of r, each appropriately left-shifted. Each copy, when shifted, is at most 2n bits long.
In this problem, we will examine a scheme for adding n binary numbers, each m bits long, using a circuit or a parallel architecture. The main parameter of interest in this question is therefore the depth of the circuit or the longest path from the input to the output of the circuit. This determines the total time taken for computing the function.
To add two m-bit binary numbers naively, we must wait for the carry bit from position i-1before we can figure out the ith bit of the answer. This leads to a circuit of depth螣(m). However, carry-lookahead circuits (seewikipedia.comif you want to know more about this) can add in螣(logn)depth.

  1. Assuming you have carry-lookahead circuits for addition, show how to add n numbers eachm bits long using a circuit of depth 螣(lognlogm).
  2. When adding three m-bit binary numbers x+y+z, there is a trick we can use to parallelize the process. Instead of carrying out the addition completely, we can re-express the result as the sum of just two binary numbersr+s, such that the ith bits of r and s can be computedindependently of the other bits. Show how this can be done. (Hint: One of the numbers represents carry bits.)
  3. Show how to use the trick from the previous part to design a circuit of depth螣(logn)for multiplying two n-bit numbers.

Compute GCD(210,588)two different ways: by finding the factorization of each number, and by using Euclid鈥檚 algorithm.
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