Understanding the Remainder Problem

The question asks for the remainder when a specific expression, \((265)^{4081} + 9\), is divided by 266. This is a classic problem involving modular arithmetic, which deals with remainders after division. We need to find the value of \((265)^{4081} + 9 \pmod{266}\).

Applying Modular Arithmetic to Calculate Remainder

To solve this efficiently, we can use the properties of modular arithmetic. The key idea is that we can find the remainder of each part of the expression first, and then combine them.

Step 1: Find the Remainder of the Base

First, let's find the remainder when the base, 265, is divided by 266.

\(265 \div 266\)

Since 265 is less than 266, the remainder is 265. In modular arithmetic, we can also express this using a negative remainder.

\(265 \equiv 265 \pmod{266}\)

Alternatively, we know that \(266 - 1 = 265\). So, 265 is 1 less than a multiple of 266 (which is \(1 \times 266\)).

\(265 \equiv -1 \pmod{266}\)

Using \(-1\) is often helpful when dealing with powers, as it simplifies calculations significantly.

Step 2: Substitute the Remainder into the Expression

Now we substitute this equivalent value (\(-1\)) back into the original expression \((265)^{4081} + 9 \pmod{266}\).

\((265)^{4081} + 9 \equiv (-1)^{4081} + 9 \pmod{266}\)

Step 3: Calculate the Power of -1

Next, we calculate \( (-1)^{4081} \). The rule is:

• \((-1)^{\text{even number}} = 1\)
• \((-1)^{\text{odd number}} = -1\)

The exponent is 4081, which is an odd number.

So, \( (-1)^{4081} = -1 \).

Step 4: Complete the Calculation Modulo 266

Substitute the value of \( (-1)^{4081} \) back into the expression:

\((-1)^{4081} + 9 \equiv -1 + 9 \pmod{266}\)

Now, perform the addition:

\(-1 + 9 = 8\)

So, the expression simplifies to:

\(8 \pmod{266}\)

The remainder when 8 is divided by 266 is simply 8, since 8 is less than 266.

Summary of Calculation Steps

Here is a quick summary of the steps taken:

1. Recognize the expression as a modular arithmetic problem: Find \((265)^{4081} + 9 \pmod{266}\).
2. Find the remainder of the base modulo the divisor: \(265 \equiv -1 \pmod{266}\).
3. Substitute the remainder: \((265)^{4081} + 9 \equiv (-1)^{4081} + 9 \pmod{266}\).
4. Calculate the power: \( (-1)^{4081} = -1 \).
5. Perform the final addition modulo the divisor: \(-1 + 9 = 8\).
6. The remainder is 8.

Final Answer

The remainder when \((265)^{4081} + 9\) is divided by 266 is 8.

Revision Table: Remainder Calculation

Additional Information: Properties of Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. Here are some key properties used in solving problems like this:

• Congruence: We write \(a \equiv b \pmod{m}\) if \(a\) and \(b\) have the same remainder when divided by \(m\). This is equivalent to saying that \(m\) divides \(a - b\).
• Addition Property: If \(a \equiv b \pmod{m}\) and \(c \equiv d \pmod{m}\), then \(a+c \equiv b+d \pmod{m}\).
• Multiplication Property: If \(a \equiv b \pmod{m}\) and \(c \equiv d \pmod{m}\), then \(a \times c \equiv b \times d \pmod{m}\).
• Exponentiation Property: If \(a \equiv b \pmod{m}\), then \(a^k \equiv b^k \pmod{m}\) for any positive integer \(k\). This property was crucial here, allowing us to replace \(265^{4081}\) with \((-1)^{4081}\) modulo 266.

Using these properties can significantly simplify calculations involving large numbers and exponents when only the remainder is required.