What is the remainder when \(3^{255}\) is divided by 28?
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:
\(3^{255} = 3^{(3 \times 85)} = (3^3)^{85}\)Calculating the Final Remainder
We can now substitute the congruence we found earlier (\(3^3 \equiv -1 \pmod{28}\)) into our expression:
\((3^3)^{85} \pmod{28} \equiv (-1)^{85} \pmod{28}\)Now, we need to calculate \((-1)^{85}\). Since 85 is an odd number, raising -1 to an odd power results in -1.
\((-1)^{85} = -1\)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:
\(-1 \pmod{28} \equiv -1 + 28 \pmod{28} \equiv 27 \pmod{28}\).Final Result
Therefore, the remainder when \(3^{255}\) is divided by 28 is 27.
Similar Questions
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Important Questions from Divisibility and Remainder
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
If all positive divisors of 132 are arranged in descending order, then what digit will be at unit place of first divisor ?
If 3 2019 is divided by 10, then what is the remainder?
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
