How to Find the Last Two Digits of a Large Power Using Modular Arithmetic

Understanding how to find the last two digits of a large power can be quite useful in various mathematical applications and competitions. This article will guide you through the process using modular arithmetic and Euler's Theorem. Let's explore how we can compute 3^{2022} mod 100 in a step-by-step manner.

Understanding the Basics

To find the last two digits of a number, we compute the remainder when this number is divided by 100. This means we need to calculate 3^{2022} mod 100. The key to solving this problem lies in leveraging concepts from number theory, specifically Euler's Theorem and the properties of modular arithmetic.

Euler's Theorem and Coprimality

Euler's Theorem states that if a and n are coprime (i.e., gcd(a, n) 1), then a^{phi(n)} equiv 1 pmod{n}, where phi(n) is Euler's totient function. For n 100, we have phi(100) 40, since phi(100) 100 cdot (1 - frac{1}{2}) cdot (1 - frac{1}{5}) 40.

Since gcd(3, 100) 1, Euler's Theorem tells us that 3^{40} equiv 1 pmod{100}. This simplifies our problem significantly as we can reduce the exponent modulo 40.

Reducing the Exponent

Given the exponent 2022, we can reduce it modulo 40:

2022 equiv 22 pmod{40}

This means that 3^{2022} equiv 3^{22} pmod{100}.

Finding 3^{22} mod 100

Now, we need to find 3^{22} mod 100. We can break this down step by step using modular exponentiation:

3^2 equiv 9 pmod{100} 3^4 (3^2)^2 equiv 9^2 81 pmod{100} 3^5 3^4 cdot 3 equiv 81 cdot 3 243 equiv 43 pmod{100} 3^{10} (3^5)^2 equiv 43^2 1849 equiv 49 pmod{100} 3^{20} (3^{10})^2 equiv 49^2 2401 equiv 1 pmod{100} 3^{22} 3^{20} cdot 3^2 equiv 1 cdot 9 9 pmod{100}

Thus, the last two digits of 3^{2022} are 09.

Conclusion

By using modular arithmetic and Euler's Theorem, we were able to simplify the problem of finding the last two digits of 3^{2022} to a more manageable calculation. This method can be applied to other similar problems by identifying the properties of the modulus and utilizing number theory theorems.

Key Takeaways:

Euler's Theorem: If gcd(a, n) 1, then a^{phi(n)} equiv 1 pmod{n}. Modular Arithmetic: Simplifies calculations by reducing larger exponents. Dividing Exponents Using Moduli: a^{bn} equiv a^{b mod phi(n)} pmod{n}. Repeated Squaring: An efficient method for modular exponentiation.

Related Keywords:

modular arithmetic Euler's Theorem last two digits large powers