If 17 2020 is divided by 18, then what is the remainder ?
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Finding the Remainder of 17 to the Power of 2020 Divided by 18
The problem asks for the remainder when a large number, specifically 17 raised to the power of 2020 (\(17^{2020}\)), is divided by 18. This type of problem can be efficiently solved using the concepts of modular arithmetic.
Understanding Modular Arithmetic for Remainders
Modular arithmetic helps us work with remainders. When we say 'a is congruent to b modulo m', written as \(a \equiv b \pmod{m}\), it means that a and b have the same remainder when divided by m. A key property is that if \(a \equiv b \pmod{m}\), then \(a^n \equiv b^n \pmod{m}\) for any positive integer n.
Applying Modular Arithmetic to the Problem
We need to find the remainder of \(17^{2020}\) when divided by 18. In modular notation, this is finding \(17^{2020} \pmod{18}\).
First, let's consider the base number, 17, in relation to the divisor, 18:
- When 17 is divided by 18, the remainder is 17. So, \(17 \equiv 17 \pmod{18}\).
- Alternatively, we can think of 17 as being just one less than 18. So, \(17 \equiv -1 \pmod{18}\). Working with -1 is often much easier, especially with exponents.
Calculating the Remainder using Congruence
Using the property of modular arithmetic for exponents, we can raise both sides of the congruence \(17 \equiv -1 \pmod{18}\) to the power of 2020:
\(17^{2020} \equiv (-1)^{2020} \pmod{18}\)
Now, we need to evaluate \((-1)^{2020}\).
- When -1 is raised to an even power, the result is 1.
- When -1 is raised to an odd power, the result is -1.
Since 2020 is an even number, \((-1)^{2020} = 1\).
Substituting this back into our congruence:
\(17^{2020} \equiv 1 \pmod{18}\)
This means that when \(17^{2020}\) is divided by 18, the remainder is 1.
Final Answer Derivation
Based on our calculation using modular arithmetic, the remainder when \(17^{2020}\) is divided by 18 is 1.
The final answer is 1.
Revision Table: Key Concepts for Remainder Problems
| Concept | Description | Application Example |
|---|---|---|
| Remainder | The amount left over after division. For \(a \div m\), remainder r means \(a = qm + r\), where \(0 \le r < m\). | 17 divided by 18 has a remainder of 17. \(17 = 0 \times 18 + 17\). |
| Modular Congruence | \(a \equiv b \pmod{m}\) means a and b have the same remainder when divided by m. Equivalently, \(a - b\) is divisible by m. | \(17 \equiv -1 \pmod{18}\) because \(17 - (-1) = 18\), which is divisible by 18. |
| Exponent Rule | If \(a \equiv b \pmod{m}\), then \(a^n \equiv b^n \pmod{m}\). | Since \(17 \equiv -1 \pmod{18}\), then \(17^{2020} \equiv (-1)^{2020} \pmod{18}\). |
Additional Information on Remainder Calculations
Finding remainders of large powers is a common type of question in number theory and competitive exams. Using modular arithmetic, especially finding a base that is congruent to 1 or -1 modulo the divisor, simplifies the calculation significantly. If the base was congruent to a number other than 1 or -1, say \(a \equiv k \pmod{m}\), we would need to calculate \(k^n \pmod{m}\). If n is large, this might involve finding a pattern in the powers of k modulo m (e.g., \(k^1, k^2, k^3, \dots \pmod{m}\)) which will eventually repeat. The length of this repeating pattern is related to Euler's totient theorem or Fermat's Little Theorem in some cases.
For this specific problem, the case of \(a \equiv -1 \pmod{m}\) is particularly simple because powers of -1 alternate between -1 and 1, depending on whether the exponent is odd or even.
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select the correct answer using the code given below:
