Divisibility Rules 4 | Number System for CAT 2026 | Ravi Prakash Rodha

Name: Divisibility Rules 4 | Number System for CAT 2026 | Ravi Prakash Rodha
Uploaded: 2018-02-28T21:00:16.000Z
Duration: 50 min 12 s

Understanding Remainders: Dividing Large Numbers

Calculating the Remainder of a Large Number by 13

The problem involves finding the remainder when the number 301302303304...400 is divided by 13. The speaker recalls that 10^3 mod 13 gives a remainder of -1, while 10^6 mod 13 gives +1.

The speaker groups the digits into sets of three to simplify calculations, noting that this large number consists of 300 digits.

Since there are an even number of groups (100), and it starts with -1, the last group will end with +1. This alternating pattern continues throughout the calculation.

Each pair of numbers results in +1, leading to a total count of pairs contributing to the final result.

After calculating, they find that dividing the total contributions (50 from pairs) by 13 yields a remainder of 11.

Transitioning to Division by 11

The next question changes focus to finding the remainder when dividing by 11 using similar logic as before.

The speaker explains that for division by 11, one can also use groups of three since 10^3 mod 11 gives -1 and alternates thereafter.

Ultimately, after performing similar calculations as with division by 13, they conclude that dividing their result (50 from earlier steps) by 11 yields a remainder of 6.

Applying Chinese Remainder Theorem for Complex Divisions

Finding Remainders for Repeated Digits

A new problem introduces a sequence "5555..." repeated until it has 602 digits and asks for its remainder when divided by 21.

Instead of directly calculating for divisibility by 21, they break it down into factors: 3 times 7, applying known rules for each factor separately.

Detailed Calculation Steps

For division by seven, they note that 10^3 mod 7 gives -1; thus grouping into threes simplifies calculations again.

After determining how many complete groups exist within those digits and what remains at the end (two fives), they calculate that this leads to a final result divisible yielding a remainder of six when divided by seven.

Final Calculation for Division by Three

Finally, they compute the sum of all digits in "5555..." repeated over its length. Summing these up leads them to find out how many times it divides evenly against three resulting in a final answer with a remainder of one.

Solving Remainder Problems Using the Chinese Remainder Theorem

Understanding the Problem

A number divided by 7 leaves a remainder of 6, indicating it can be expressed as 7x + 6 .

The same number when divided by 3 leaves a remainder of 1, leading to the expression 3y + 1 .

Finding Solutions

To find the smallest number satisfying both conditions, one can either use trial and error or apply elimination techniques from the Chinese Remainder Theorem (CRT) .

By substituting values for x, starting with x = 0 and x = 1, we determine that at x = 1, y = 4 is an integer solution .

Final Answer

The smallest number that meets both conditions is determined to be 13. This showcases a practical application of the Chinese Remainder Theorem .

Calculating Remainders for Large Numbers

Introduction to the Problem

A large repeating number sequence "5647" is analyzed to find its remainder when divided by 101. The task involves understanding how powers of ten behave under modulo operations .

Methodology Overview

It’s established that 10^2 mod 101 equiv -1, while 10^4 mod 101 equiv +1. This allows grouping digits in pairs or fours for simplification .

Grouping Strategy

Two methods are proposed:

Method One: Grouping in pairs leads to alternating remainders of plus and minus based on previous calculations.

Method Two: Grouping in fours consistently yields a positive remainder due to properties of powers of ten modulated by 101 .

Detailed Calculation Steps

In Method One, groups yield results where every two contribute negatively or positively based on their arrangement. Ultimately, this leads to calculating total contributions from all groups .

Conclusion on Results

Both methods converge on the same final answer: 32, demonstrating consistency across different approaches in modular arithmetic [].

Exploring Modular Arithmetic with Different Bases

Flexibility in Grouping Techniques

Discussion emphasizes flexibility in choosing group sizes based on specific numbers like 11 and 9, which have unique properties regarding their remainders when using powers of ten .

Practical Applications

For example, using groups of two with base 11 shows no negative signs appear due to consistent positive remainders from even groupings. This highlights strategic choices in problem-solving within modular arithmetic contexts .

This structured approach provides clarity on solving complex mathematical problems involving remainders and modular arithmetic through systematic methodologies such as CRT and strategic grouping techniques.