"Mathematical Algorithms | Number Digits" is a guide that explores problems and solutions involving digits. It covers common digit-related issues, such as digit properties, manipulation, and counting under constraints. The article discusses algorithmic approaches like modular arithmetic, string handling, recursion, and math libraries. It also highlights applications in areas like cryptography, verification, error detection, and combinatorics. The article concludes with practice problems for readers to reinforce their understanding.

Common Problems:

• Digit-based Properties: Algorithms can identify numbers with specific digit properties. Examples include:
  • Automorphic Numbers: A number whose square ends with the original number itself (e.g., 5^2 = 25).
  • Self-Descriptive Numbers: The digits of the number represent its base raised to the power of those digits (e.g., 21 = 2^1 * 1^2).
  • Harshad Numbers (Niven Numbers): Divisible by the sum of their digits (e.g., 18 is divisible by 1 + 8).
• Digit Manipulation: Algorithms can perform operations on number digits:
  • K-th Digit of a^b: Efficiently finding the k-th digit of a large power (e.g., 2nd digit of 7^100).
  • Building Lowest Number: Removing n digits from a number to form the smallest possible number.
  • Next Greater Number with Same Digits: Rearranging digits of a number to form the next greater number.
• Counting and Generation: Algorithms can enumerate or generate numbers based on digit constraints:
  • Number of n-digit Stepping Numbers: Numbers where consecutive digits differ by 1 (e.g., 123, 432).
  • Numbers with Unique Digits: Counting or generating numbers within a range with no repeated digits.

Algorithmic Approaches:

These problems often involve techniques like:

• Modular Arithmetic: Performing calculations with remainders to exploit digit-level properties.
• String Manipulation: Treating numbers as strings of digits for digit-specific operations.
• Recursion: Efficiently solving problems for larger digits by breaking them down into smaller subproblems.
• Math Libraries: Utilizing built-in libraries for mathematical functions like exponentiation.

Applications:

Algorithms dealing with number digits have various applications in computer science:

• Cryptography: Certain number properties can be used to design cryptographic systems.
• Verification and Error Detection: Algorithms can validate data integrity by checking digit-based checksums (e.g., ISBN codes).
• Combinatorics: Counting or generating specific arrangements of digits can be relevant in probability and optimization problems.

Practice Problems on Number Digit:

• Automorphic Number
• Self Descriptive Number
• Motzkin number
• Narcissistic number
• Schröder–Hipparchus number
• Number of n digit stepping numbers
• Deserium Number
• K’th Boom Number
• Pandigital Product
• Neon Number
• Harshad (Or Niven) Number
• Tidy Number (Digits in non-decreasing Order)
• Seeds (Or Seed Roots) of a number
• Numbers having Unique (or Distinct) digits
• Build Lowest Number by Removing n digits from a given number
• n-th number with digits in {0, 1, 2, 3, 4, 5}
• Number of Digits in a^b
• Brahmagupta Fibonacci Identity
• First digit in factorial of a number
• Number with even sum of digits
• Largest even digit number not greater than N
• Number of digits in 2 raised to power n
• n-th number whose sum of digits is ten
• Minimum digits to remove to make a number Perfect Square
• Count digits in given number N which divide N
• Count digit groupings of a number with given constraints
• Print first k digits of 1/n where n is a positive integer
• Program to check if a given number is Lucky
• Check if a given number can be represented in given a no. of digits in any base
• Find element using minimum segments in Seven Segment Display
• Find nth term of the Dragon Curve Sequence
• Find the Largest Cube formed by Deleting minimum Digits from a number
• Find next greater number with same set of digits
• Find the Number which contain the digit d
• Find nth number that contains the digit k or divisible by k
• Find N integers with given difference between product and sum
• Find the number of consecutive zero at the end after multiplying n numbers
• Find first and last digits of a number
• Find a number x such that sum of x and its digits is equal to given n
• Find smallest permutation of given number
• Find maximum number that can be formed using digits of a given number
• Find count of digits in a number that divide the number
• Find the occurrences of digit d in the range [0..n]
• Find sum of digits in factorial of a number
• Find n-th element in a series with only 2 digits (4 and 7) allowed
• First digit in product of an array of numbers
• Find n’th number in a number system with only 3 and 4
• Find the smallest number whose digits multiply to a given number n
• Find the Largest number with given number of digits and sum of digits
• Form the largest number using at most one swap operation
• Form a number using corner digits of powers
• Position of n among the numbers made of 2, 3, 5 & 7
• Powers of 2 to required sum
• Expressing a number as sum of consecutive | Set 2 (Using odd factors)
• Sum of digits written in different bases from 2 to n-1
• Given a number N in decimal base, find number of its digits in any base (base b)
• Reverse a number using stack
• Check if a number is jumbled or not
• Number of digits in the product of two numbers
• Form the smallest number using at most one swap operation
• Difference between sums of odd and even digits
• Numbers having difference with digit sum more than s
• Count n digit numbers not having a particular digit
• Program to check Plus Perfect Number
• Total numbers with no repeated digits in a range
• K-th digit in ‘a’ raised to power ‘b’
• Possible to make a divisible by 3 number using all digits in an array
• Recursive sum of digit in n^x, where n and x are very large
• Check if all digits of a number divide it
• Counting numbers whose difference from reverse is a product of k
• Maximize a number considering permutations with values smaller than limit
• Number of decimal numbers of length k, that are strict monotone
• Count square and non-square numbers before n
• Counting numbers of n digits that are montone
• Count numbers in range L-R that are divisible by all of its non-zero digits
• Round the given number to nearest multiple of 10
• First occurrence of a digit in a given fraction
• Check if a number can be written as sum of three consecutive integers
• Count numbers with unit digit k in given range
• Convert to number with digits as 3 and 8 only
• Find a Number X whose sum with its digits is equal to N
• Count different numbers that can be generated such that there digits sum is equal to ‘n’
• Check whether a number can be represented by sum of two squares
• Total no of 1’s in numbers
• Largest number with prime digits
• Largest smaller number possible using only one swap operation
• Representation of a number in powers of other
• Sum of digits of a given number to a given power
• Number of n digit numbers that do not contain 9
• Check if given number can be represented as sum of two great numbers
• Biggest integer which has maximum digit sum in range from 1 to n
• Largest number less than N whose each digit is prime number
• Smallest x such that 1*n, 2*n, … x*n have all digits from 1 to 9
• Count of numbers satisfying m + sum(m) + sum(sum(m)) = N
• Next higher palindromic number using the same set of digits
• Program to count digits in an integer (3 Different Methods)
• Next higher number using atmost one swap operation
• Smallest number k such that the product of digits of k is equal to n
• Count number of digits after decimal on dividing a number
• Find n-th element in a series with only 2 digits (4 and 7) allowed | Set 2 (log(n) method)
• To check a number is palindrome or not without using any extra space
• Minimum number with digits as 4 and 7 only and given sum
• Number of times a number can be replaced by the sum of its digits until it only contains one digit
• Print last k digits of a^b (a raised to power b)
• Number of digits in the nth number made of given four digits
• Count total number of digits from 1 to n
• Count numbers with difference between number and its digit sum greater than specific value
• Given a number n, find the first k digits of n^n
• Generate k digit numbers with digits in strictly increasing order
• Pandigital number in a given base
• Reverse digits of an integer with overflow handled
• Number of occurrences of 2 as a digit in numbers from 0 to n
• Digital Root (repeated digital sum) of the given large integer
• Largest number smaller than or equal to n and digits in non-decreasing order
• Check if frequency of each digit is less than the digit
• N’th palindrome of K digits
• Maximum and minimum sums from two numbers with digit replacements
• Last non-zero digit of a factorial
• Count numbers with same first and last digits
• Generate all palindromic numbers less than n
• Minimum sum of two numbers formed from digits of an array
• Smallest number by rearranging digits of a given number
• Largest palindrome which is product of two n-digit numbers
• Count ways to spell a number with repeated digits
• Finding sum of digits of a number until sum becomes single digit
• Finding number of digits in n’th Fibonacci number
• Reverse and Add Function
• Print all n-digit numbers whose sum of digits equals to given sum
• Program to find last digit of n’th Fibonnaci Number
• Count positive integers with 0 as a digit and maximum ‘d’ digits
• Count ‘d’ digit positive integers with 0 as a digit
• Count numbers having 0 as a digit
• Check if a given number is Fancy
• Find the most frequent digit without using array/string
• Count total number of N digit numbers such that the difference between sum of even and odd digits is 1
• Count of n digit numbers whose sum of digits equals to given sum
• Total number of non-decreasing numbers with n digits
• Count numbers from 1 to n that have 4 as a digit
• Compute sum of digits in all numbers from 1 to n
• Remove repeated digits in a given number
• Replace all ‘0’ with ‘5’ in an input Integer
• Count numbers that don’t contain 3
• Write a C program to reverse digits of a number