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Chapter 1: Q15E (page 49)

Determine necessary and sufficient conditions on xandc so that the following holds: for anya,b, if ax鈮�bxmodc, thena鈮�bmodc .

Short Answer

Expert verified

The necessary and sufficient condition forxandc is(a鈭�b) must be divisible byc as theGCD(c,x) is equal to1 .

Step by step solution

01

Introduction

Fermat鈥檚 Little Theorem states that for any prime number xand integer i, the number i.e. ix鈥�iis the factor of x.

02

Given condition

The given is,

For anya,b,c

If ax=bx鈥�modc

To prove:a=bmodc

03

Condition we can define from the given statement 

Ifax=bx鈥�modc

Then,

c(a鈭�b)x

If a=bmodc,

c(a鈭�b)

04

Explanation

So, here c divides (a鈭�b)x then, c must divide (a鈭�b).

For the value of x, theGCD(c,x) must equal to 1 that ensure that(a鈭�b)x is divisible by c.

Thus, in this case(a鈭�b) must be divisible byc as theGCD(c,x) is equal to 1.

Therefore, the necessary and sufficient condition forxandc is(a鈭�b) must be divisible byc as theGCD(c,x) is equal to 1.

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Most popular questions from this chapter

Show that if a鈮�b(modN)and if M Divides Nthena鈮�b(modM)

Is the difference of 530,000and6123,456a multiple of31 ?

Starting from the definition of x鈮�ymodN(namely, that Ndivides x-y), justify the substitution rule x鈮�x'modN,y鈮�y'modNx+y鈮�x'+y'modN,and also the corresponding rule for multiplication.

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.

How many integers modulo113 have inverses?(Note:113=1331)
See all solutions

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