Question

Find the remainder of \(5^{1258}\) when divided by 12 .

Short answer

The remainder of \(5^{1258}\) when divided by 12 is 1.

Step-by-step solution

Understanding Modulo Properties

To find the remainder when a large power is divided by a number, we can use modular arithmetic. Specifically, Fermat's Little Theorem simplifies computations when dealing with powers. According to Fermat's Little Theorem, if a is not divisible by a prime p, then \(a^{p-1} \equiv 1 \pmod{p}\). For this problem, we should first check if this theorem applies or consider simplifying the expression differently.

Identify the Suitable Modulo Theorem

Since 12 is not a prime, Fermat’s Little Theorem cannot be directly applied as it is. However, we should see if smaller moduli, such as 4 and 3 (whose product is 12), can help by using the Chinese Remainder Theorem. We will compute two simpler expressions: \(5^{1258} \mod 4\) and \(5^{1258} \mod 3\). Then, we combine the results.

Compute \( 5^{1258} \mod 4 \)

First, notice that \(5 \equiv 1 \pmod{4}\). Thus, \(5^n \equiv 1^n \equiv 1 \pmod{4}\) for any positive integer \(n\). Therefore, \(5^{1258} \equiv 1 \pmod{4}\).

Compute \( 5^{1258} \mod 3 \)

Now, let's evaluate \(5^{1258} \mod 3\). Notice that \(5 \equiv 2 \pmod{3}\). Using the property that \(2^2 \equiv 1 \pmod{3}\), we see that \(2^n\) repeats every two powers modulo 3: \(2^1 \equiv 2 \pmod{3}\) and \(2^2 \equiv 1 \pmod{3}\). Since 1258 is even, \(5^{1258} = 2^{1258} = (2^2)^{629} \equiv 1^{629} \equiv 1 \pmod{3}\).

Apply Chinese Remainder Theorem

Now, having both results: \(5^{1258} \equiv 1 \pmod{4}\) and \(5^{1258} \equiv 1 \pmod{3}\), notice that a number \(x\) satisfies both congruences modulo 3 and 4 if and only if it is equivalent to 1 modulo 12 (since 12 = 3 x 4). Thus \(5^{1258} \equiv 1 \pmod{12}\).

Key concepts

Modular Arithmetic

Modular arithmetic is an essential part of number theory and is sometimes described as "clock arithmetic." It deals with integers where numbers "wrap around" upon reaching a certain value called the modulus. For example, if you think about a clock, after reaching 12, you start again at 1. This is similar to saying 13 ≡ 1 (mod 12), where 12 is the modulus. When doing math with a modulus, you're essentially figuring out the remainder after division by that modulus. This makes calculations with very large numbers manageable.

• Helps in simplifying calculations by focusing on remainders.
• Useful in various real-world applications like computer science, cryptography, and digital signal processing.
• Key operator: '≡', which means "is congruent to."

Fermat's Little Theorem

Fermat's Little Theorem is a fundamental theorem in number theory that is especially useful in modular arithmetic. It states that if you have a prime number \( p \), and an integer \( a \) that is not divisible by \( p \), then \( a^{p-1} \equiv 1 \pmod{p} \). This theorem provides a way to reduce large exponential expressions modulo a prime into something much simpler.While it is incredible for simplifying problems involving primes, it can't be used directly with non-prime moduli like 12. Instead, it is useful to find relationships and shortcuts for computations with prime factors of other numbers through the use of other strategies such as the Chinese Remainder Theorem.

• Simplifies large powers in modular arithmetic.
• Requires the modulus to be a prime number.
• Often a stepping stone to solving broader modular problems.

Congruences

The concept of congruences is central to modular arithmetic. Congruences express that two numbers have the same remainder when divided by a modulus. For instance, saying \( a \equiv b \pmod{m} \) means when \( a \) is divided by \( m \), it leaves the same remainder as \( b \). This allows us to classify numbers into equivalence classes where they behave the same under division by \( m \).Understanding congruences helps us solve complex problems more easily, as you'll often break down larger numbers into equivalent forms that are much simpler to work with. Through congruences, a massive exponential expression can boil down to an easy-to-handle number.

• Foundation for modular equations.
• Helps in solving equations by looking only at remainders.
• Works with both prime and composite moduli.

Remainder Computation

Computing remainders is a fundamental operation in both mathematics and everyday life. Approaches like modular arithmetic allow mathematicians to focus specifically on the remainder rather than the full quotient when dividing one integer by another.The goal in problems like finding the remainder of \( 5^{1258} \div 12 \) is to simplify the calculations by recognizing patterns of powers modulo smaller integers. This often involves reducing the problem to finding equivalent expressions that ultimately yield a remainder through several steps, as we applied dividing the power over smaller moduli like 3 and 4, then recombining these results using tools like the Chinese Remainder Theorem.

• Simplifies calculations, especially with large numbers.
• Involves breaking down problems and using pattern recognition.
• Can use the Chinese Remainder Theorem for integration of smaller modulus results.