Try it on GfG Practice Lucas numbers are similar to Fibonacci numbers. Lucas numbers are also defined as the sum of its two immediately previous terms. But here the first two terms are 2 and 1 whereas in Fibonacci numbers the first two terms are 0 and 1 respectively. Mathematically, Lucas Numbers…

Try it on GfG Practice The digital root of a positive integer is found by summing the digits of the integer. If the resulting value is a single digit then that digit is the digital root. If the resulting value contains two or more digits, those digits are summed and the process is repeated. This is…

Given a series 2, 12, 36, 80, 150.. Find the n-th term of the series. Examples : Input : 2 Output : 12 Input : 4 Output : 80 If we take a closer look, we can notice that series is sum of squares and cubes of natural numbers (1, 4, 9, 16, 25, .....) + (1, 8, 27, 64, 125, ....). Therefore n-th number…

Given the number of terms i.e. n. Find the sum of the series 0.6, 0.06, 0.006, 0.0006, ...to n terms. Examples: Input: 2 Output: 0.66 Explanation: sum of the series upto 2 terms: 0.6 + 0.06 = 0.66. Input: 3 Output: 0.666 Explanation: sum of the series upto 3 terms: 0.6 + 0.06 + 0.006 = 0.666. Table…

Given a series 12 + 32 + 52 + 72 + . . . + (2*n - 1)2, find the sum of the series. Examples: Input: n = 4 Output: 84 Explanation: sum = 12 + 32 + 52 + 72 = 1 + 9 + 25 + 49 = 84 Input: n = 10 Output: 1330 Explanation: sum = 12 + 32 + 52 + 72 + 92 + 112 + 132 + 152 + 172 + 192 = 1 + 9 + 24 + 49 + . .…

Given an integer n. The task is to find the sum of the following sequence: 2, 22, 222, ......... to n terms. Examples : Input: n = 2 Output: 24 Explanation: For n = 2, the sum of first 2 terms are 2 + 22 = 24 Input: 3 Output: 246 Explanation: For n = 3, the sum of first 3 terms are 2 + 22 + 222 =…

Try it on GfG Practice Given n, count all 'a' and 'b' that satisfy the condition a^3 + b^3 = n. Where (a, b) and (b, a) are considered two different pairs Examples: Input: n = 9 Output: 2 Explanation: 1^3 + 2^3 = 9 and 2^3 + 1^3 = 9 Input: n = 28 Output: 2 Explanation: 1^3 + 3^3 = 28 and 3^3 + 1^3 =…

Try it on GfG Practice Padovan Sequence similar to Fibonacci sequence with similar recursive structure. The recursive formula is, P(n) = P(n-2) + P(n-3) P(0) = P(1) = P(2) = 1 Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...... Spiral of squares with side lengths which follow the Fibonacci…

Try it on GfG Practice In this article, we will learn how to find the LCM of given array elements. Given an array of n numbers, find the LCM of it. Example: Input : {1, 2, 8, 3} Output : 24 LCM of 1, 2, 8 and 3 is 24 Input : {2, 7, 3, 9, 4} Output : 252 Table of Content [Naive Approach] Iterative…

Try it on GfG Practice Given a number n, find the last non-zero digit in n!.Examples: Input : n = 5 Output : 2 5! = 5 * 4 * 3 * 2 * 1 = 120 Last non-zero digit in 120 is 2. Input : n = 33 Output : 8 Recommended Practice Last non-zero digit in factorial Try It! A Simple Solution is to first find n!,…

Try it on GfG Practice Given two positive integers k and n, where k > 1, find the largest power of k that divides n! (n factorial). Examples: Input: n = 7, k = 2 Output: 4 Explanation: 7! = 5040, and 24 = 16 is the highest power of 2 that divides 5040. Input: n = 10, k = 9 Output: 2 Explanation: 10!…

Try it on GfG Practice Factorial of a non-negative integer, is the multiplication of all integers smaller than or equal to n. Factorial of a number Examples: Input: 100 Output: 933262154439441526816992388562667004- 907159682643816214685929638952175999- 932299156089414639761565182862536979-…

Try it on GfG Practice Given a positive integer n ( 1 <= n <= 1015). Find the largest prime factor of a number. Input: 6 Output: 3 Explanation Prime factor are 2 and 3. Largest of them is 3. Input: 15 Output: 5 Explanation: Prime factors are 3 and 5. The largest of them is 5. Input: 28 Output: 7…

Try it on GfG Practice Given an integer n, find all unique prime factors of n. A prime factor is a prime number that divides n exactly (without leaving a remainder). Examples: Input: n = 100 Output: [2, 5] Explanation: Unique prime factors of 100 are 2 and 5. Input: n = 60 Output: [2, 3, 5]…

Try it on GfG Practice Given a positive integer n, find all the distinct divisors of n. Examples: Input: n = 10 Output: [1, 2, 5, 10] Explanation: 1, 2, 5 and 10 are the divisors of 10. Input: n = 100 Output: [1, 2, 4, 5, 10, 20, 25, 50, 100] Explanation: 1, 2, 4, 5, 10, 20, 25, 50 and 100 are…

Try it on GfG Practice Given an integer n, find the value of Euler's Totient Function, denoted as Φ(n). The function Φ(n) represents the count of positive integers less than or equal to n that are relatively prime to n. Euler's Totient function Φ(n) for an input n is the count of numbers in {1, 2,…

Try it on GfG Practice Given a number n, write code to find the sum of digits in the factorial of the number. Given n ≤ 5000 Examples: Input : 10 Output : 27 Input : 100 Output : 648 Recommended Practice Sum of digits in factorial of a number Try It! It is not possible to store a number as large as…

Try it on GfG Practice Given a number n, print all primes smaller than n. Input: N = 10 Output: 2, 3, 5, 7 Explanation : The output “2, 3, 5, 7” for input N = 10 represents the list of the prime numbers less than or equal to 10. Input: N = 5 Output: 2, 3, 5 Explanation : The output “2, 3, 5” for…

Try it on GfG Practice Given two numbers n and m, find the n-th root of m. In mathematics, the n-th root of a number m is a real number that, when raised to the power of n, gives m. If no such real number exists, return -1. Examples: Input: n = 2, m = 9 Output: 3 Explanation: 32 = 9 Input: n = 3, m…

Try it on GfG Practice Given two numbers n and k, print k-th prime factor among all prime factors of n. For example, if the input number is 15 and k is 2, then output should be "5". And if the k is 3, then output should be "-1" (there are less than k prime factors). Examples: Input : n = 225, k = 2…

You are given a m liter jug and a n liter jug where 0 < m < n. Both the jugs are initially empty. The jugs don’t have markings to allow measuring smaller quantities. You have to use the jugs to measure d liters of water where d < n. Determine the minimum no of operations to be performed to obtain d…

Try it on GfG Practice There are N people standing in a circle waiting to be executed. The counting out begins at some point in the circle and proceeds around the circle in a fixed direction. In each step, a certain number of people are skipped and the next person is executed. The elimination…

Try it on GfG Practice Given a number ‘n’, write a function that prints the last two digits of n-th (‘n’ can also be a large number) Fibonacci number. Examples: Input : n = 65 Output : 65 Input : n = 365 Output : 65 Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.…

Try it on GfG Practice Given a value n, find the n'th even Fibonacci Number. Examples : Input n = 3 Output 34 Explanation The first 3 even Fibonacci numbers are 0, 2, 8, 34, 144, with the 3rd being 34. Input n = 4 Output 144 Explanation The first 4 even Fibonacci numbers are 0, 2, 8, 34, 144, with…

Starting with any positive integer N, Collatz sequence is defined corresponding to n as the numbers formed by the following operations : If n is even, then n = n / 2.If n is odd, then n = 3*n + 1.Repeat above steps, until it becomes 1. Examples : Input : 3 Output : 3, 10, 5, 16, 8, 4, 2, 1 Input : 6…

Try it on GfG Practice A number n is said to be a Carmichael number if it satisfies the following modular arithmetic condition: power(b, n-1) MOD n = 1, for all b ranging from 1 to n such that b and n are relatively prime, i.e, gcd(b, n) = 1 Given a positive integer n, find if it is a Carmichael…

Given two polynomials represented by two arrays, write a function that adds given two polynomials. Example: Input: A[] = {5, 0, 10, 6} B[] = {1, 2, 4} Output: sum[] = {6, 2, 14, 6} The first input array represents "5 + 0x^1 + 10x^2 + 6x^3" The second array represents "1 + 2x^1 + 4x^2" And Output is…

Try it on GfG Practice Power Set P(S) of a set S is the set of all subsets of S. For example S = {a, b, c} then P(s) = {{}, {a}, {b}, {c}, {a,b}, {a, c}, {b, c}, {a, b, c}}. If S has n elements in it then P(s) will have 2n elements Examples Input: s = "ab" Output: "", "a", "b", "ab" Explanation: The…

Try it on GfG Practice Given a 32-bit integer n and an integer d, rotate the binary representation of n by d positions in both left and right directions. After each rotation, convert the result back to its decimal representation and return both values in an array as [left rotation, right rotation].…

Try it on GfG Practice Given a number x and two positions (0 based from the right side) in the binary representation of x, write a function that swaps n bits at the given two positions and returns the result. It is also given that the two sets of bits do not overlap. Examples: Input: x = 47…

Try it on GfG Practice Write an efficient program to count the number of 1s in the binary representation of an integer. Examples : Input : n = 6 Output : 2 Binary representation of 6 is 110 and has 2 set bits Input : n = 13 Output : 3 Binary representation of 13 is 1101 and has 3 set bits [Naive…

Try it on GfG Practice Given an integer n, the task is to return the position of the first set bit from right to left in the binary representation n. If n is equal to 0 (no set bits present), then return 0. Note: Position of rightmost bit is 1. Examples: Input: n = 18 Output: 2 Explanation: Binary…

Given a number, find the greatest number less than the given a number which is the power of two or find the most significant bit number . Examples: Input: 10 Output: 8 Explanation:Greatest number which is a Power of 2 less than 10 is 8 Binary representation of 10 is 1010 The most significant bit…

Given two integers a and b, the task is to multiply them without using the multiplication operator. Instead of that, use the Russian Peasant Algorithm. Examples: Input: a = 2, b = 5 Output: 10 Explanation: Product of 2 and 5 is 10. Input: a = 6, b = 9 Output: 54 Explanation: Product of 6 and 9 is…

Try it on GfG Practice Given two variables a and y, swap two variables without using a third variable. Examples: Input: a = 2, b = 3 Output: a = 3, b = 2 Input: a = 20, b = 0 Output: a = 0, b = 20 Input: a = 10, b = 10 Output: a = 10, b = 10 Table of Content Using Arithmetic Operators Store the sum…

Given two integers a and b, the task is to determine whether they have opposite signs. Return true if the signs of the two numbers are different and false otherwise. Examples: Input: a = -5, b = 10 Output: true Explanation: One number is negative and the other is positive, so their signs are…

Given two numbers, the task is to check if two numbers are equal without using Arithmetic and Comparison Operators or String functions. Method 1 : The idea is to use XOR operator. XOR of two numbers is 0 if the numbers are the same, otherwise non-zero. C++ // C++ program to check if two numbers //…

Given two integers, the task is to find XOR of them without using the XOR operator. Examples : Input: x = 1, y = 2 Output: 3 Input: x = 3, y = 5 Output: 6 Approach - Checking each bit - O(log n) time and O(1) space A Simple Solution is to traverse all bits one by one. For every pair of bits, check…

Try it on GfG Practice Given a number n containing only 1 set bit in its binary representation, the task is to find the position of the only set bit. If there are 0 or more than 1 set bits, then return -1. Note: Position of set bit '1' should be counted starting with 1 from the LSB side in the…

Try it on GfG Practice Given a positive integer n, the task is to find if it is a power of 2 or not. Examples: Input : n = 16 Output : Yes Explanation: 24 = 16 Input : n = 42 Output : No Explanation: 42 is not a power of 2 Input : n = 1 Output : Yes Explanation: 20 = 1 Approach 1: Using Log - O(1)…