Question

Find the remainder of \(9^{1573}\) when divided by 11 .

Short answer

The remainder is 3.

Step-by-step solution

Identify the Problem Type

The given problem requires us to find the remainder of dividing a large exponentiation problem by a number, which suggests using Fermat's Little Theorem. This theorem helps simplify calculations of powers modulo a prime.

Recall Fermat’s Little Theorem

Fermat's Little Theorem states that if p is a prime number and a is an integer not divisible by p, then \(a^{p-1} \equiv 1 \pmod{p}\). Since 11 is a prime, apply this theorem to simplify the problem.

Apply the Theorem

According to Fermat's Little Theorem, \(9^{10} \equiv 1 \pmod{11}\), since 11 is prime and 9 is not divisible by 11.

Use the Theorem for Large Powers

Instead of calculating \(9^{1573}\) directly, express it as \(9^{1573} = 9^{10 \cdot 157 + 3}\). By Fermat’s theorem, \((9^{10})^{157} \equiv 1^{157} \equiv 1 \pmod{11}\). This simplifies our expression to finding \(9^3 \mod 11\).

Calculate Small Powers

Now, calculate \(9^3\). First, calculate \(9^2 = 81\). Then, multiply by 9 again: \(9^3 = 81 \times 9 = 729\).

Determine Final Remainder

Find the remainder when 729 is divided by 11. Perform the division: \(729 \div 11 \approx 66.27\). Thus, \(729 = 11 \times 66 + 3\), so the remainder is 3.

Key concepts

Understanding Modular Arithmetic

Modular arithmetic is an essential tool in number theory, often referred to as "clock arithmetic." It deals with remainders and cycles back around when a threshold, known as the modulus, is reached. Instead of focusing on the actual values, modular arithmetic focuses on their remainders after division.

Here's a simple analogy: Think of a 12-hour clock. If it's 7:00 now, in 5 hours, it will be back at 12:00 rather than 19:00 since clocks "reset" every 12 hours. In modulo terms, you'd say \(7 + 5 \equiv 0 \mod 12\).Modular arithmetic is beneficial when working with large numbers. It helps break down complex calculations into simpler versions that are far easier to handle.

In the context of the problem at hand, we are interested in the remainder when large powers, like \(9^{1573}\), are divided by 11. This can seem overwhelming without the tools of modular arithmetic to simplify the problem.

Insight into Prime Number Theorem

The prime number theorem is a key component in number theory, stating that prime numbers become less frequent as numbers grow larger, but they are infinite and have special properties. Fermat's Little Theorem, which is a result that also hinges on the concept of prime numbers, allows us to simplify calculations involving divisions and remainders.

• Prime numbers have unique divisibility traits. They are only divisible by 1 and themselves.
• This property is integral for Fermat’s Little Theorem, as it provides the foundation for the theorem's reliability.
• Since 11 is a prime number, the theorem is applicable, greatly simplifying what could otherwise be an unwieldy exponentiation task.

With 11 being prime, Fermat’s Little Theorem tells us that any integer's power up to 10 (11 - 1) modulo 11 will cycle around predictably. This predictability is harnessed in the problem where knowing the power sequence of 9 modulo 10 simplifies the calculation of large exponents.

Performing Remainder Calculations

Performing remainder calculations is a skillful act in number theory, often requiring division followed by determining the leftover part or remainder. This procedure might seem straightforward, but when handling large numbers, this can become complex without the right approach.

Once expanding \(9^{1573}\) is reduced to just finding \(9^{3} \mod 11\), through the application of Fermat's Theorem, the remaining steps are a breeze. Start by calculating smaller exponents, as large exponent numbers are decomposed through the theorem:

- Compute \(9^2 = 81\) - Then \(9^3 = 81 \times 9 = 729\).Finally, the division of 729 by 11 gives a quotient of approximately 66, leaving a remainder of 3. Thus, \(729 \equiv 3 \mod 11\).

This demonstrated how solving modular equations often involves breaking complex calculations into series of simpler mathematical operations.

Simplifying Large Exponents

Simplifying large exponents often leverages the power of modular arithmetic and mathematical theorems like Fermat's Little Theorem. Attempting to compute or even understand gigantic powers directly, such as \(9^{1573}\), is impractical, but by using the tools provided by number theory, it becomes accessible.

• Fermat’s Little Theorem simplifies exponentiation when dealing with prime moduli by reducing the exponentiation cycle.
• The theorem states only the remainder of cycles matters after a certain threshold when involving primes, meaning calculations can halt significantly sooner.

For example, in this exercise, we applied the theorem to rewrite \(9^{1573}\) as \(9^{10 \cdot 157 + 3}\) due to the property \(9^{10} \equiv 1 \mod 11\).

The inherent property lets us focus solely on the power of 3 instead of the mammoth 1573, demonstrating again the efficacy of these tools in tackling even the most challenging numerical problems.