Mathematical Algorithms – Divisibility and Large Numbers

Divisibility is about whether one number can be divided by another number without leaving any remainder. This is an important concept in math. In this article, we’ll look at the different ways to test if a number is divisible by another number. We’ll explain the techniques used to quickly figure out if a number can be divided evenly, without any leftovers. Understanding divisibility helps us work with large numbers more efficiently.

Divisibility plays a crucial role in number theory, providing a foundation for understanding relationships between numbers. An algorithm is a set of well-defined instructions used to solve a problem or complete a task. In the context of divisibility, algorithms help us determine whether one number is divisible by another.

One of the most fundamental algorithms for divisibility is the division algorithm. This algorithm states that for any two positive integers a and b, there exist unique integers q and r such that:

a = bq + r

where 0 ≤ r < b.

Here, q is the quotient, r is the remainder, and b is the divisor. If the remainder r is 0, then a is divisible by b. This algorithm forms the basis for many other divisibility tests, such as:

• Divisibility by 2: A number is divisible by 2 if its last digit is even.
• Divisibility by 3: The sum of the digits of a number is divisible by 3 if the number itself is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if the last two digits are divisible by 4.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.

These tests provide quick and efficient ways to determine divisibility without performing actual division.

Large numbers, often exceeding the capacity of our everyday counting systems, require specialized algorithms for efficient manipulation. These algorithms are crucial in various fields, including cryptography, scientific computing, and financial modeling.

One prominent algorithm for handling large numbers is the Karatsuba multiplication algorithm . This algorithm offers a more efficient way to multiply large numbers compared to the traditional grade-school multiplication method. It works by recursively dividing the numbers into smaller parts, multiplying those parts, and then combining the results.

Another important algorithm is the Fast Fourier Transform (FFT) . This algorithm is used to efficiently compute the Discrete Fourier Transform (DFT), which is a mathematical operation that decomposes a signal into its constituent frequencies. The FFT finds applications in signal processing, image compression, and data analysis.

• Check if a large number is divisible by 3 or not
• Number of digits to be removed to make a number divisible by 3
• Find whether a given integer is a power of 3 or not
• Check if a large number is divisible by 4 or not
• Count rotations divisible by 4
• Number of substrings divisible by 4 in a string of integers
• Check if a large number is divisible by 6 or not
• Prove that atleast one of three consecutive even numbers is divisible by 6
• Sum of all numbers divisible by 6 in a given range
• Number of substrings divisible by 6 in a string of integers
• Print digit’s position to be removed to make a number divisible by 6
• Check divisibility by 7
• To check whether a large number is divisible by 7
• Remainder with 7 for large numbers
• Count rotations divisible by 8
• Given a large number, check if a subsequence of digits is divisible by 8
• Check if a large number is divisible by 9 or not
• Decimal representation of given binary string is divisible by 10 or not
• Check if a large number is divisible by 11 or not
• Program to find remainder when large number is divided by 11
• Divisibility by 12 for a large number
• Check if a large number is divisible by 13 or not
• Check if a large number is divisibility by 15
• Check if a large number is divisible by 20
• Number is divisible by 29 or not
• Check if a larger number divisible by 36
• Divisible by 37 for large numbers
• To check divisibility of any large number by 999
• Sub-string Divisibility by 3 Queries
• Sub-string Divisibility by 11 Queries
• Queries for counts of multiples in an array
• Check if a number is multiple of 5 without using / and % operators
• Smallest number divisible by first n numbers
• Divide a big number into two parts that differ by k
• Sum of n digit numbers divisible by a given number
• Count n digit numbers divisible by given number
• Find set of m-elements with difference of any two elements is divisible by k
• Largest number by which given 3 numbers should be divided such that they leaves same remainder
• Difference of two large numbers
• Sum of two large numbers
• Find Last Digit Of a^b for Large Numbers
• Divide large number represented as string
• Multiply Large Numbers represented as Strings
• Writing power function for large numbers
• Recursive sum of digit in n^x, where n and x are very large
• Check whether given floating point number is even or odd
• Smallest n digit number divisible by given three numbers
• Largest number that divides x and is co-prime with y
• Program to compute division upto n decimal places
• Find element in array that divides all array elements
• Program for quotient and remainder of big number
• Given a HUGE number check if it’s a power of two
• Print k numbers where all pairs are divisible by m
• 9’s complement of a decimal number
• Divide number into two parts divisible by given numbers
• Smallest K digit number divisible by X
• Largest K digit number divisible by X
• Rearrangement of a number which is also divisible by it
• Find the number closest to n and divisible by m
• Divisibility by 3 where each digit is the sum of all prefix digits modulo 10
• Count the numbers divisible by ‘M’ in a given range
• Number of Groups of Sizes Two Or Three Divisible By 3
• Find if a number is divisible by every number in a list
• Count of m digit integers that are divisible by an integer n
• Biggest number by arranging numbers in certain order
• Large Fibonacci Numbers in Java
• Nth Square free number
• Refactorable number
• Find the largest multiple of 3 from array of digits | Set 2 (In O(n) time and O(1) space)
• Partition a number into two divisble parts
• Count natural numbers whose factorials are divisible by x but not y
• Compute (a*b)%c such that (a%c) * (b%c) can be beyond range