Understanding the Remainder Problem

The question asks us to find the remainder when the large number \(3^{255}\) is divided by 28. In mathematical terms, we need to calculate \(3^{255} \pmod{28}\).

Applying Modular Arithmetic

Modular arithmetic is a way of working with remainders. The expression '\(a \equiv b \pmod{m}\)' means that 'a' and 'b' have the same remainder when divided by 'm'. We can use properties of modular arithmetic to simplify calculations involving large exponents.

Let's find the remainders of the first few powers of 3 when divided by 28:

• \(3^1 \div 28\) has a remainder of 3. So, \(3^1 \equiv 3 \pmod{28}\).
• \(3^2 = 9\). \(9 \div 28\) has a remainder of 9. So, \(3^2 \equiv 9 \pmod{28}\).
• \(3^3 = 27\). \(27 \div 28\) has a remainder of 27. So, \(3^3 \equiv 27 \pmod{28}\).

Notice that 27 is very close to 28. We can write this using negative remainders as well: \(27 \equiv 27 - 28 \pmod{28}\), which means \(27 \equiv -1 \pmod{28}\). This simplification is very helpful!

Simplifying the Exponent \(3^{255}\)

We found that \(3^3 \equiv -1 \pmod{28}\). Now let's look at the exponent, 255. We can rewrite 255 using the power 3:

Divide 255 by 3: \(255 \div 3 = 85\).

So, we can express the exponent 255 as \(3 \times 85\).

Now we can rewrite \(3^{255}\) using this:

Calculating the Final Remainder

We can now substitute the congruence we found earlier (\(3^3 \equiv -1 \pmod{28}\)) into our expression:

Now, we need to calculate \((-1)^{85}\). Since 85 is an odd number, raising -1 to an odd power results in -1.

So, \(3^{255} \equiv -1 \pmod{28}\).

A remainder must be a non-negative number less than the divisor (28). To convert -1 to the standard remainder:

Final Result

Therefore, the remainder when \(3^{255}\) is divided by 28 is 27.