What day of the week will it be a googolplex days from now?

Therefore, in googolplex days it will be a Thursday. 12. Suppose that p and q are distinct primes, ap ≡ a (mod q), and aq ≡ a (mod p).

What is the remainder of 2432 when divided by 11 using Fermat’s little theorem?

4
Note that Fermat’s Little Theorem tells us that 210 ⌘ 1 mod 10, which means that we can replace 210 in this equation with 1. So we have 2432 = 243022 = (210)4322 ⌘ 14322 ⌘ 1 · 22 ⌘ 4 (mod 1)1. Hence, the remainder of dividing 2432 by 11 is 4.

Why Fermat’s little theorem is useful?

Fermat’s little theorem is a fundamental theorem in elementary number theory, which helps compute powers of integers modulo prime numbers. It is a special case of Euler’s theorem, and is important in applications of elementary number theory, including primality testing and public-key cryptography.

Is Fermat’s little theorem proved?

Fermat’s “biggest”, and also his “last” theorem states that xn + yn = zn has no solutions in positive integers x, y, z with n > 2. This has finally been proven by Wiles in 1995.

What is the remainder when to power 125 divided by 11?

So the remainder is 5.

How do you find a remainder on a cat?

Remainder obtained when N is divided by 5 is -2, then the positive remainder is given by 5-3 = 2. Similarly, if 4 is the positive remainder obtained when some number is divided by 7, then the negative remainder is given by 4-7 = -3.

How does Fermat’s little theorem work?

Take an Example How Fermat’s little theorem works. Examples: P = an integer Prime number a = an integer which is not multiple of P Let a = 2 and P = 17 According to Fermat’s little theorem 2 17 – 1 ≡ 1 mod (17) we got 65536 % 17 ≡ 1 that mean (65536-1) is an multiple of 17. Use of Fermat’s little theorem.

How to find the value of φ(n)?

Knowing only n, the computation of φ(n) has essentially the same difficulty as the factorization of n, since φ(n) = (p − 1) (q − 1), and conversely, the factors p and q are the (integer) solutions of the equation x2 – (n − φ(n) + 1) x + n = 0 .

What are the generalizations of Euler’s and Fermat’s theorem?

Generalizations. A corollary of Euler’s theorem is: for every positive integer n, if the integer a is coprime with n then for any integers x and y . This follows from Euler’s theorem, since, if , then for some integer k, and one has If n is prime, this is also a corollary of Fermat’s little theorem.

What is the Lucas primality test for prime numbers?

This theorem forms the basis for the Lucas primality test, an important primality test, and Pratt’s primality certificate . If a and p are coprime numbers such that ap−1 − 1 is divisible by p, then p need not be prime. If it is not, then p is called a (Fermat) pseudoprime to base a.