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106 LearnersLast updated on September 11, 2025
Fermat's Little Theorem Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like number theory. Whether you’re exploring prime numbers, cryptography, or modular arithmetic, calculators will make your life easy. In this topic, we are going to talk about Fermat's Little Theorem calculators.
What is Fermat's Little Theorem Calculator?
A Fermat's Little Theorem calculator is a tool to determine results in modular arithmetic using Fermat's Little Theorem.
This theorem states that if p is a prime number and a is an integer not divisible by p , then ap-1 ≡ 1p mod(p) .
This calculator helps perform these calculations much easier and faster, saving time and effort.
How to Use the Fermat's Little Theorem Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the base a and the prime p : Input the integers into the given fields.
Step 2: Click on calculate: Click on the calculate button to compute ap-1 ≡ 1p mod(p) and get the result.
Step 3: View the result: The calculator will display the result instantly.
How Does Fermat's Little Theorem Work?
Fermat's Little Theorem is used to simplify calculations in modular arithmetic.
It states that for a prime p and an integer a such that p does not divide a : ap-1 ≡ 1p mod(p)
This theorem is particularly useful in computing large powers modulo a number, especially in cryptography.
Tips and Tricks for Using the Fermat's Little Theorem Calculator
When we use a Fermat's Little Theorem calculator, there are a few tips and tricks that we can use to make it a bit easier and avoid mistakes:
Ensure that p is a prime number; otherwise, the theorem does not apply.
Remember that a should not be divisible by p for the theorem to work.
Use the theorem to simplify calculations of large powers in modular arithmetic.
Common Mistakes and How to Avoid Them When Using the Fermat's Little Theorem Calculator
We may think that when using a calculator, mistakes will not happen. But it is possible for children to make mistakes when using a calculator.
Mistake 1
Using a non-prime number for p
Ensure that the value of p is a prime number. If p is not prime, the theorem does not hold true.
Mistake 2
Incorrectly selecting a divisible by p
The theorem only applies when a is not divisible by p . Double-check that a and p are coprime.
Mistake 3
Misinterpreting the modulo operation
Understanding modular arithmetic is crucial as it affects the outcome. Make sure to properly compute ap-1 ≡ 1p mod(p)
Mistake 4
Relying on the calculator for all number theory problems
While Fermat's Little Theorem simplifies specific computations, it's not applicable to all number theory scenarios. Understand its scope and limitations.
Mistake 5
Assuming the calculator handles all modular arithmetic
Not all modular arithmetic problems can be solved with this theorem. Ensure that you use the theorem's conditions properly.
Fermat's Little Theorem Calculator Examples
Problem 1
Calculate \( 7^{16} \mod 17 \).
Use Fermat's Little Theorem:
Since 17 is prime, 716 ≡1 pmod (17).
Thus, 716 mod 17 = 1 .
Explanation
According to Fermat's Little Theorem, for ap-1 ≡ 1p mod(p) p prime and a not divisible by p .
Problem 2
Determine \( 3^{22} \mod 23 \).
Use Fermat's Little Theorem:
Since 23 is prime, 322 ≡1 pmod23.
Thus, 322 mod 23 = 1.
Explanation
Fermat's Little Theorem simplifies the calculation to 1 since 23 is prime and 3 is not divisible by 23.
Problem 3
Find \( 5^{10} \mod 11 \).
Use Fermat's Little Theorem:
Since 11 is prime, 510 ≡ 1 pmod(11).
Thus, 510 mod 11 = 1.
Explanation
The theorem applies as 11 is prime and 5 is not divisible by 11.
Problem 4
Compute \( 2^{100} \mod 101 \).
Use Fermat's Little Theorem:
Since 101 is prime, 2100 ≡1 pmod(101).
Thus, 2100 mod 101 = 1.
Explanation
According to Fermat's Little Theorem, a prime p and a not divisible by p satisfy the theorem.
Problem 5
Evaluate \( 9^{14} \mod 13 \).
Use Fermat's Little Theorem:
Since 13 is prime, 912 ≡1 pmod(13).
Thus, 914 = 912 × 92 ≡ 1 × 81 ≡ 3 pmod(13).
Explanation
Using the theorem and simplifying powers, we find the result is 3 modulo 13.
FAQs on Using the Fermat's Little Theorem Calculator
1.How do you apply Fermat's Little Theorem?
2.What happens if \( a \) is divisible by \( p \)?
3.Why is Fermat's Little Theorem useful?
4.How do I use a Fermat's Little Theorem calculator?
5.Is the Fermat's Little Theorem calculator always accurate?
Glossary of Terms for the Fermat's Little Theorem Calculator
- Fermat's Little Theorem: A principle in number theory that helps simplify calculations in modular arithmetic for prime moduli.
- Modular Arithmetic: A system of arithmetic for integers where numbers wrap around upon reaching a certain value, the modulus.
- Prime Number: A natural number greater than 1 that has no divisors other than 1 and itself.
- Coprime: Two numbers that have no common divisors other than 1.
- Modulo Operation: A mathematical operation that returns the remainder after division of one number by another.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
