Number System || Finding Remainder ? (LESSON-4)

Summary

This video explains the process of finding remainders when large powers of numbers are divided by specific divisors, with a focus on division by 5. The presenter demonstrates how to break down powers into manageable components using modular arithmetic, recognizing patterns to simplify the calculations. Key techniques such as checking for remainders and dividing powers into smaller chunks are covered in detail. The speaker emphasizes the importance of practice and consistency in mastering the technique for competitive exams, ensuring that learners can efficiently solve modular problems and avoid confusion with negative remainders.

Takeaways

• 😀 The key to solving modular arithmetic problems is understanding the cycles of remainders for powers of numbers.
• 😀 To find remainders, divide the base number by the divisor and note the remainder for smaller powers of the base.
• 😀 Identifying patterns or cycles in powers of numbers helps simplify large calculations, especially with remainders.
• 😀 In modular arithmetic, breaking down exponents allows for easier calculation of remainders without fully expanding the powers.
• 😀 The process involves converting powers of numbers into smaller, manageable forms and using known cycles of remainders.
• 😀 For example, when dividing 2^517 by 5, we use the pattern of powers of 2 mod 5 to find the remainder is 2.
• 😀 In the case of 3^165 divided by 5, recognizing the cycles of powers of 3 modulo 5 simplifies the process to yield a remainder of 3.
• 😀 Using modular arithmetic, once you break down the problem into smaller parts, you can often cancel out large numbers and reduce the problem to a simple remainder.
• 😀 The tutorial encourages solving many practice problems to build fluency in recognizing patterns and applying the methods correctly.
• 😀 Understanding the logic behind finding remainders using modular arithmetic makes even complex problems much easier to solve.

Q & A

What is the first step in solving the problems involving finding the remainder of powers of numbers?

The first step is to divide the individual number by the divisor (like 5, 4, etc.) and find the remainder. This helps simplify the overall calculation.

How do you approach problems where the remainder is not immediately obvious?

You need to find the nearest multiples of the divisor, which helps in simplifying the powers of numbers and determining a manageable pattern for remainders.

What was the main challenge in the example where 517 is divided by 5, and how was it solved?

The challenge was identifying the remainder when 517 is divided by 5. After dividing, the remainder was found to be 2. This allowed the equation to be simplified as 2^5, and further steps showed the remainder to be 2.

What method did the speaker use to simplify the calculation of 2^517 divided by 5?

The speaker divided the large number (517) by 5 to find the remainder. Then, they looked for nearby multiples of 5, leading to a simplified calculation involving powers of 2 and modulo arithmetic.

In the example of dividing 517 by 5, how did the speaker handle the large numbers?

The speaker broke down the number 517 into manageable parts by finding the remainder after dividing by 5, simplifying the equation step by step, and using the modular arithmetic approach to identify the remainder.

What was the result when the number 517 was divided by 4, and how did it affect the final answer?

When 517 was divided by 4, the quotient was 129 with a remainder of 1. This remainder allowed the speaker to simplify the expression further and ultimately find the remainder as 2 when divided by 5.

Why did the speaker choose 2^4 (or 16) as the base in the division by 5?

The speaker chose 2^4 (16) because it is close to a multiple of 5, making it easier to calculate the remainder when divided by 5.

What is the significance of using powers of 3 when finding the remainder in the example involving 3^165?

The powers of 3 were used because the speaker needed to find a remainder that could easily be handled by dividing by 5. Powers of 3 were chosen based on their proximity to multiples of 5, making calculations easier.

What strategy did the speaker use to solve the division of 3^165 by 4 and then by 5?

The speaker divided 3^165 by 4 to simplify the equation, yielding a quotient and remainder. Then, the calculation was reduced by finding the remainder when powers of 3 were divided by 5, using modular arithmetic.

What was the final remainder when 3^165 was divided by 5, and how was it determined?

The final remainder when 3^165 was divided by 5 was 3. This was determined by simplifying 3^165 to a smaller power of 3 and dividing by 5, ultimately finding that the remainder is 3.