Finding the Remainder of \((17^{25} + 19^{25})\) Divided by 18

This solution explains how to find the remainder when the expression \((17^{25} + 19^{25})\) is divided by \(18\). We will use the principles of modular arithmetic to simplify the calculation.

Understanding the Problem

We need to calculate the value of \(R\) in the equation:

where \(R\) is the remainder and must be \(0 \le R < 18\).

Applying Modular Arithmetic

Modular arithmetic allows us to work with remainders. The core idea is that if \(a \equiv b \pmod{m}\), then \(a^n \equiv b^n \pmod{m}\) for any positive integer \(n\). We first find the remainders of the bases (\(17\) and \(19\)) when divided by \(18\).

• Remainder of \(17\) divided by \(18\):

  \(17 \div 18\) gives a quotient of \(0\) and a remainder of \(17\). In modular arithmetic, we can also express this using a negative remainder, which is often simpler:

  \(17 \equiv 17 \pmod{18}\)

  Alternatively, since \(17 = 18 - 1\), we can write:

  \(17 \equiv -1 \pmod{18}\)
• Remainder of \(19\) divided by \(18\):

  \(19 \div 18\) gives a quotient of \(1\) and a remainder of \(1\).

  \(19 \equiv 1 \pmod{18}\)

Step-by-Step Calculation

Now, we substitute these congruences back into the original expression:

Using the properties of modular arithmetic, we replace \(17\) with \(-1\) and \(19\) with \(1\):

Next, we evaluate the powers:

• \((-1)^{25}\): Since the exponent \(25\) is an odd number, \((-1)^{25} = -1\).
• \(1^{25}\): Any positive integer power of \(1\) is \(1\). So, \(1^{25} = 1\).

Substitute these results back into the expression:

Calculate the sum:

Final Remainder

The result of the calculation is \(0 \pmod{18}\). This means that when \((17^{25} + 19^{25})\) is divided by \(18\), the remainder is \(0\).