Question

The least positive remainder when \(17^{30}\) is divided by 5 is (a) 2 (b) 4 (c) 3 (d) 1

Short answer

The least positive remainder when \(17^{30}\) is divided by 5 is (b) 4.

Step-by-step solution

Find the remainder when the base is divided by 5

We want to find the remainder when 17 is divided by 5. We have: \[17 \equiv 2 \pmod 5\]

Raise the remainder to the power of 30

Now, we raise the remainder obtained in step 1 (which is 2) to the power of 30: \[2^{30}\] Note that we are not directly computing the value of \(2^{30}\), but rather, finding the remainder when it is divided by 5.

Determine the pattern of powers of 2 modulo 5

To find the remainder when \(2^{30}\) is divided by 5, we start by observing the powers of 2 modulo 5: \(2^1 \equiv 2 \pmod{5}\), \(2^2 \equiv 4 \pmod{5}\), \(2^3 \equiv 3 \pmod{5}\), \(2^4 \equiv 1 \pmod{5}\) We notice that there is a pattern that repeats every 4 powers, i.e., \((\)2, 4, 3, 1)$.

Find the remainder in the pattern of powers

Since we have discovered in step 3 that the powers of 2 modulo 5 repeat every 4 powers, we can find the remainder when \(2^{30}\) is divided by 4: \[30 \div 4 = 7 \text{ with a remainder of } 2\] Using the remainder (2), we know that it corresponds to the second number in the \((\)2, 4, 3, 1)$ pattern. Therefore: \[2^{30} \equiv 4 \pmod 5\]

Answer

Based on our calculations, the least positive remainder when \(17^{30}\) is divided by 5 is 4. Hence, the correct answer is: (b) 4

Key concepts

Remainder Theorem

The remainder theorem is a powerful tool in modular arithmetic that simplifies calculations involving division. In essence, the theorem allows us to replace a number with its remainder when divided by a given modulus. This concept becomes especially useful when dealing with large numbers raised to high powers, such as in the exercise provided.

• Understanding remainders: When a number is divided by another, the remainder is what is left over. For example, when 17 is divided by 5, we are left with a remainder of 2, since 17 can be expressed as 5 times 3, plus 2.
• Application: In modular arithmetic, instead of using the original number, we use its remainder. This greatly simplifies calculations, particularly those involving exponentiation.

For our exercise, applying the remainder theorem allowed us to simplify 17 to 2 (i.e., modulo 5), which made further calculations with powers much easier.

Patterns in Powers

In modular arithmetic, recognizing patterns in the powers of a base number can significantly ease calculations. Observing these patterns can reveal cyclical behavior, allowing us to predict results without direct computation of large powers. Understanding these patterns was key to solving the exercise.

• Cycle discovery: We explored the powers of 2 modulo 5, resulting in a repetitive sequence: 2, 4, 3, and 1. This cycle repeats every four powers.
• Predicting outcomes: Once we identify such a cycle, we can use it to predict the result of very high powers. For example, calculating 2 raised to any power greater than 4 just involves finding where it lies in the cycle.

In the exercise, 2 raised to the 30th power was determined by locating the position of 30 within the four-number pattern, landing on 4 as the remainder.

Cyclicity in Modular Systems

Cyclicity in modular arithmetic refers to the repeated cycles that occur when calculating powers of numbers modulo some number. Observing these cycles can vastly simplify arithmetic operations, allowing mathematicians to solve problems more efficiently and correctly.

• Identifying cycles: Through the powers of 2 modulo 5, we discovered a cycle every four steps. Understanding where these cycles start and stop is crucial for predicting results accurately without laborious calculations.
• Using cycles in solving problems: To solve modular problems, align the power of interest with the cycle. In the exercise, the 30th power was reduced to 2 based on how many times the cycle of four patterns is repeated, and where it stands within the cycle.

This concept of cyclicity permits us to efficiently manage otherwise complex calculations in modular arithmetic.