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Chapter 10: Problem 28

Solve the following modular equations. a. \([8]_{10} x=[4]_{10}\) b. \([4]_{17} x=[5]_{17}\)

Short Answer

Expert verified
The solutions to the modular equations are: a) x=8, b) x=5

Step by step solution

01

Solve the first modular equation

For the equation \( [8]_{10} x = [4]_{10} \), the goal is to find an x, such that when x is multiplied by 8 and then the result is divided by 10, the remainder is 4. One can begin by setting x equal to 1 and then incrementally increasing the value of x and checking if the condition is satisfied. Using this method, the problem can be solved and the value of x can be found to be 8.
02

Solve the second modular equation

One can solve the equation \( [4]_{17} x = [5]_{17} \) using the same method as in Step 1. One simply sets x equal to 1 and then incrementally increase the value of x and checks if the condition is satisfied. Using this method, the problem can be solved and the value of x can be found to be 5.

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Most popular questions from this chapter

Show that if \(q\) is a factor of \(n\) and \(k\) is the order of \(q\) in \(n,\) then \(q^{k} | B(n, q),\) where \(B(n, q)\) denotes the binomial coefficient.

Consider an RSA cryptosystem using \(p=23, q=\) 41 and \(g=3 .\) Encipher the message \([847]_{943}\).

Prove that if \(s \in[m]_{n}\) and \(t \in[k]_{n}\) then \(s \times t \in[m\) \(\times k] n\).

Prove that if \(m\) and \(n\) are both even, then gcd \((m, n)=2 \operatorname{gcd}(m / 2, n / 2)\).

If an integer between 1 and 10,000 is randomly chosen according to the uniform distribution, approximately what is the probability of it being prime?
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