Solution to x^k=b mod m Using Prime Factors and Mod Congruence

Thread starter lifom
Start date Mar 30, 2011

In summary, the equation "x^k=b mod m" involves finding the value of x that, when raised to the power of k and divided by m, has a remainder of b. This is a type of modular arithmetic problem, where operations are performed on remainders rather than traditional numbers. The possible solutions to this equation depend on the values of k, b, and m, and can range from no solutions to infinitely many solutions. To solve this equation, one can use methods such as the Chinese Remainder Theorem or modular exponentiation. In real life, this equation has many applications in cryptography, computer science, engineering, and mathematics.

Mar 30, 2011

lifom

Let m be a product of distinct primes p1,p2,...pr.
Assume x=c (mod pi) is a solution of x^k=b (mod pi) for all i =1,2,3,...r.
Can I conclude that x=c (mod m) is a solution of x^k=b (mod m) ?

(I think that if a=b mod x and a=b mod y then a=b mod(xy), provided that gcd(x,y)=1)

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Mar 30, 2011

robert2734

lifom said:

Can I conclude that x=c (mod m) is a solution of x^k=b (mod m) ?

Yes.

FAQ: Solution to x^k=b mod m Using Prime Factors and Mod Congruence

What is the meaning of "x^k=b mod m" in this equation?

In this equation, x represents the base, k represents the exponent, b represents the remainder, and m represents the modulus. The equation is asking for the value of x that, when raised to the power of k and divided by m, has a remainder of b.

How is this equation related to modular arithmetic?

The equation "x^k=b mod m" is an example of a modular arithmetic problem. Modular arithmetic deals with operations on remainders, rather than traditional numbers. In this equation, the modulus m determines the range of possible remainders, and the remainder b is the specific value that we are looking for.

What are the possible solutions for this equation?

There can be multiple solutions for this equation, depending on the values of k, b, and m. In some cases, there may be no solutions. In other cases, there may be infinitely many solutions. It all depends on the relationship between k, b, and m.

How can this equation be solved?

There are various methods for solving equations of this form, depending on the values of k, b, and m. One approach is to use the Chinese Remainder Theorem, which is a formula for finding the value of x when given the remainders and moduli of a set of congruences. Another approach is to use modular exponentiation, which involves repeatedly squaring the base and reducing the result modulo m until the desired exponent is reached.

What are the applications of this equation in real life?

Modular arithmetic, and therefore equations like "x^k=b mod m", have many practical applications. One example is in cryptography, where modular arithmetic is used to encrypt and decrypt data. It is also used in computer science for tasks such as generating random numbers and checking for data errors. In the field of engineering, modular arithmetic is used in data compression and error correction. It also has applications in coding theory, number theory, and other areas of mathematics.