Calculating Remainders Using Fermat's Little Theorem
Fermat's Little Theorem is a powerful tool in number theory that simplifies the process of calculating remainders, especially when working with large exponents. This theorem can be used to find the remainder when a number is divided by another number, making it an essential concept for understanding and solving various problems in modular arithmetic.
Introduction to Fermat's Little Theorem
Fermat's Little Theorem states that if ( p ) is a prime number, and ( a ) is any integer not divisible by ( p ), then:
[ a^{p-1} equiv 1 pmod{p} ]For example, in the problem of finding the remainder when ( 2^{100} ) is divided by ( 101 ), we can apply this theorem to simplify the calculation.
Applying Fermat's Little Theorem to Find the Remainder
Method 1: Direct Application of Fermat's Little Theorem
Given:
( p 101 ) (a prime number)
( a 2 ) (an integer not divisible by 101)
Using Fermat's Little Theorem:
[ 2^{100} equiv 1 pmod{101} ]Therefore, the remainder when ( 2^{100} ) is divided by 101 is 1. This is due to the theorem's direct application and simplifies the calculation significantly.
Method 2: Detailed Step-by-Step Calculation
We can also verify the result by breaking down the problem step-by-step:
[ 2^{10} 1024 equiv 14 pmod{101} ]
[ 2^{100} 14^{10} ]
Next, we need to simplify ( 14^{10} pmod{101} ) in smaller steps:
[ 7^2 49 equiv 49 pmod{101} ]
[ 7^3 343 equiv 343 - 3 times 101 40 pmod{101} ]
[ 7^5 7^3 times 49 40 times 49 equiv 1960 equiv 2020 - 60 -60 pmod{101} ]
[ 7^{10} (-60)^2 3600 equiv 3600 - 35 times 101 3600 - 3535 65 pmod{101} ]
[ 2^{100} 65 times 14 910 equiv 9091 equiv 1 pmod{101} ]
Hence, the remainder when ( 2^{100} ) is divided by 101 is 1.
Applications of Fermat's Little Theorem
The theorem has broad applications in various fields, including cryptography, where it is used in the RSA algorithm for secure data transmission. It also simplifies the process of calculating large powers in modular arithmetic, making it a valuable tool in number theory and beyond.
Conclusion
Fermat's Little Theorem provides a straightforward method for finding remainders when dividing by a prime number. Whether using a direct application of the theorem or breaking down the problem into smaller steps, the remainder when ( 2^{100} ) is divided by 101 is consistently found to be 1.
Understanding and applying Fermat's Little Theorem enhances problem-solving skills in modular arithmetic and prepares one for more advanced concepts in number theory and cryptography.