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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 n=1,  21=2(mod3)For n=2,  22=1(mod3)For n=3,  23=2(mod3)For n=4,  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

1.38. To see if a number, say 562437487, is divisible by 3, you just add up the digits of its decimalrepresentation, and see if the result is divisible by role="math" localid="1658402816137" 3.

( 5+6+2+4+3+7+4+8+7=46, so it is not divisible by 3).

To see if the same number is divisible by 11, you can do this: subdivide the number into pairs ofdigits, from the right-hand end(87,74,43,62,5) , add these numbers and see if the sum is divisible by11 (if it's too big, repeat).

How about 37? To see if the number is divisible by 37, subdivide it into triples from the end(487,437,562) add these up, and see if the sum is divisible by37 .


This is true for any prime pother than2 and 5. That is, for any prime p≠2,5, there is an integer rsuch that in order to see ifp divides a decimal number n, we breakn into r-tuples of decimal digits (starting from the right-hand end), add up these r-tuples, and check if the sum is divisible by p.

(a) What is the smallest rsuch for p=13? Forp=13 ?

(b) Show thatr is a divisor of p-1.

In the RSA cryptosystem, Alice’s public key (N,e)is available to everyone. Suppose that her private key d is compromised and becomes known to Eve. Show that e=3if (a common choice) then Eve can efficiently factor N.

Prove that the grade-school multiplication algorithm (page 24), when applied to binary numbers, always gives the right answer.

Suppose you want to compute the nth Fibonacci number Fn , modulo an integer p. Can you find an efficient way to do this?

Suppose that instead of using a compositeN=pqin the RSA cryptosystem (Figure 1.9), we simply use a prime modulus p . As in RSA, we would have an encryption exponent e, and the encryption of a message mmodpwould be memodp.Prove that this new cryptosystem is not secure, by giving an efficient algorithm to decrypt: that is, an algorithm that given and p,e,andmemodp as input, computes . Justify the correctness and analyze the running time of your decryption algorithm.
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