, For example, the unique factorization of the Gaussian integers is convenient in deriving formulae for all Pythagorean triples and in proving Fermat's theorem on sums of two squares. are distributed as shown in the following table (Wagon 1991). Number Theory - Euclid's Algorithm - Stanford University Thus there are infinitely many solutions, and they are given by, Later, we shall often wish to solve \(1 = x p + y q\) for coprime integers \(p\) In such a field with m numbers, every nonzero element a has a unique modular multiplicative inverse, a1 such that aa1=a1a1modm. This inverse can be found by solving the congruence equation ax1modm,[69] or the equivalent linear Diophantine equation[70], This equation can be solved by the Euclidean algorithm, as described above. given in Book VII of Euclid's Elements. cannot be infinite, so the algorithm must eventually fail to produce the next step; but the division algorithm can always proceed to the (N+1)th step provided rN > 0. Euclidean division reduces all the steps between two exchanges into a single step, which is thus more efficient. et al. The fact that the GCD can always be expressed in this way is known as Bzout's identity. The [132] The algorithm is unlikely to stop, since almost all ratios a/b of two real numbers are irrational. R1 R2 = Q3 remainder R3. 3 the largest integer that leaves a remainder zero for all numbers.. HCF of 12, 15 is 3 the largest number which exactly divides all the numbers i . number theory - Calculating RSA private exponent when given public Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps. Therefore, the greatest common divisor g must divide rN1, which implies that grN1. The Euclidean algorithm, also called Euclid's algorithm, is an algorithm for finding the greatest common divisor of two numbers a and b. Since each prime p divides L by assumption, it must also divide one of the q factors; since each q is prime as well, it must be that p=q. Iteratively dividing by the p factors shows that each p has an equal counterpart q; the two prime factorizations are identical except for their order. Since the remainders are non-negative integers that decrease with every step, the sequence given integers \(a, b, c\) find all integers \(x, y\) such that. In the late 5th century, the Indian mathematician and astronomer Aryabhata described the algorithm as the "pulverizer",[34] perhaps because of its effectiveness in solving Diophantine equations. n = m = gcd = . python Share Art of Computer Programming, Vol. The Euclidean algorithm, and thus Bezout's identity, can be generalized to the context of Euclidean domains. Before we present a formal description of the extended Euclidean algorithm, let's work our way through an example to illustrate the main ideas. r [139] By defining an analog of the Euclidean algorithm, Gaussian integers can be shown to be uniquely factorizable, by the argument above. [147][148] The basic principle is that each step of the algorithm reduces f inexorably; hence, if f can be reduced only a finite number of times, the algorithm must stop in a finite number of steps. The convergent mk/nk is the best rational number approximation to a/b with denominator nk:[134], Polynomials in a single variable x can be added, multiplied and factored into irreducible polynomials, which are the analogs of the prime numbers for integers. 66 12 = 5 remainder 6 Thus, 66 12 you will have quotient 5 and remainder 6, Step 3: Since the remainder isnt zero continue the process and you will get the result as follows. First, divide the larger number by the smaller number. 4. The extended Euclidean algorithm uses the same framework, but there is a bit more bookkeeping. To begin, multiples of 462 are subtracted from 1071 until the remainder is less than 462. Each quotient polynomial is chosen such that each remainder is either zero or has a degree that is smaller than the degree of its predecessor: deg[rk(x)] < deg[rk1(x)]. The algorithm is based on the below facts. 344 and 353-357). 78 66 = 1 remainder 12 The binary GCD algorithm is an efficient alternative that substitutes division with faster operations by exploiting the binary representation used by computers. Unique factorization is essential to many proofs of number theory. when the algorithm is applied to two consecutive Fibonacci numbers. The set of all integral linear combinations of a and b is actually the same as the set of all multiples of g (mg, where m is an integer). where a, b and c are given integers. is a Euclidean algorithm (Inkeri 1947, Barnes and Swinnerton-Dyer 1952). Lam showed that the number of steps needed to arrive at the greatest common divisor for two numbers less than is, where Lastly. The obvious answer is to list all the divisors \(a\) and \(b\), Therefore, every common divisor of and is a common divisor of and , so the procedure can be iterated as follows: For integers, the algorithm terminates when divides exactly, at which point corresponds to the greatest This may be seen by multiplying Bzout's identity by m. Therefore, the set of all numbers ua+vb is equivalent to the set of multiples m of g. In other words, the set of all possible sums of integer multiples of two numbers (a and b) is equivalent to the set of multiples of gcd(a, b). The maximum numbers of steps for a given , We first attempt to tile the rectangle using bb square tiles; however, this leaves an r0b residual rectangle untiled, where r0Euclidean Algorithm / GCD in Python - Stack Overflow and A051012). By dividing both sides by c/g, the equation can be reduced to Bezout's identity. The Euclidean Algorithm for finding GCD (A,B) is as follows: If A = 0 then GCD (A,B)=B, since the GCD (0,B)=B, and we can stop. In the initial step k=0, the remainders are set to r2 = a and r1 = b, the numbers for which the GCD is sought. A concise Wolfram Language implementation Euclids algorithm is a very efficient method for finding the GCF. The extended Euclidean algorithm is particularly useful when a and b are coprime (or gcd is 1). [45], The Euclidean algorithm was the first integer relation algorithm, which is a method for finding integer relations between commensurate real numbers. where This principle relies on the well-ordering property of the non-negative integers, which asserts that every non-empty set of non-negative integers has a smallest member.[149]. , be the number of divisions required to compute using the Euclidean algorithm, and define if . The GCD is said to be the generator of the ideal of a and b. But if we replace \(t\) with any integer, \(x'\) and \(y'\) still satisfy After that rk and rk1 are exchanged and the process is iterated. The GCD is calculated according to the Euclidean algorithm: 195 = (1)154 + 41 195 = ( 1) 154 + 41. The process of substituting remainders by formulae involving their predecessors can be continued until the original numbers a and b are reached: After all the remainders r0, r1, etc. In the given numbers 66 is small so divide 78 with it. [25][29] The algorithm may even pre-date Eudoxus,[30][31] judging from the use of the technical term (anthyphairesis, reciprocal subtraction) in works by Euclid and Aristotle. [90] In this case the total time for all of the steps of the algorithm can be analyzed using a telescoping series, showing that it is also O(h2). The solution is to combine the multiple equations into a single linear Diophantine equation with a much larger modulus M that is the product of all the individual moduli mi, and define Mi as, Thus, each Mi is the product of all the moduli except mi. Therefore, 12 is the GCD of 24 and 60. This tau average grows smoothly with a[100][101], with the residual error being of order a(1/6) + , where is infinitesimal. If B = 0 then GCD (A,B)=A, since the GCD (A,0)=A, and we can stop. of the Ferguson-Forcade algorithm (Ferguson The The step b:= a mod b is equivalent to the above recursion formula rk rk2 mod rk1. He holds several degrees and certifications. At the end of the loop iteration, the variable b holds the remainder rk, whereas the variable a holds its predecessor, rk1. obtain a crude bound for the number of steps required by observing that if we Three multiples can be subtracted (q1=3), leaving a remainder of 21: Then multiples of 21 are subtracted from 147 until the remainder is less than 21. Find GCD of 72 and 54 by listing out the factors. Since the number of steps N grows linearly with h, the running time is bounded by. The algorithm for rational numbers was Let \(d = \gcd(a,b)\), and let \(b = b'd, a = a'd\). Online calculator: Extended Euclidean algorithm - PLANETCALC [113] This is exploited in the binary version of Euclid's algorithm. big o - Time complexity of Euclid's Algorithm - Stack Overflow Below is an implementation of the above approach: Time Complexity: O(log N)Auxiliary Space: O(log N). On the other hand, it has been shown that the quotients are very likely to be small integers. First the Greatest Common Factor of the two numbers is determined from Euclid's algorithm. Let g = gcd(a,b). If gcd(a,b)=1, then a and b are said to be coprime (or relatively prime). In this case, the above becomes, \[ 3 = 27 - 4\times(33 - 1\times 27) = (-4)\times 33 + 5\times 27) \], \[ x = k m + t b / d , y = k n + t a /d .\]. 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If it does, the fraction a/b is a rational number, i.e., the ratio of two integers, and can be written as a finite continued fraction [q0; q1, q2, , qN]. Step 1: find prime factorization of each number: Step 1: Place the numbers inside division bar: Step 3: Continue to divide until the numbers do not have a common factor. of the general case to the reader. Seven multiples can be subtracted (q2=7), leaving no remainder: Since the last remainder is zero, the algorithm ends with 21 as the greatest common divisor of 1071 and 462. (y1 (b/a).x1) = gcd (2), After comparing coefficients of a and b in (1) and(2), we get following,x = y1 b/a * x1y = x1. ( is the totient function, gives the average number [150] In other words, a greatest common divisor may exist (for all pairs of elements in a domain), although it may not be possible to find it using a Euclidean algorithm. [42] Lejeune Dirichlet's lectures on number theory were edited and extended by Richard Dedekind, who used Euclid's algorithm to study algebraic integers, a new general type of number. First, we divide the bigger Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. A step of the Euclidean algorithm that replaces the first of the two numbers corresponds to a step in the tree from a node to its right child, and a step that replaces the second of the two numbers corresponds to a step in the tree from a node to its left child. This restriction on the acceptable solutions allows some systems of Diophantine equations with more unknowns than equations to have a finite number of solutions;[68] this is impossible for a system of linear equations when the solutions can be any real number (see Underdetermined system). The fundamental theorem of arithmetic applies to any Euclidean domain: Any number from a Euclidean domain can be factored uniquely into irreducible elements. [82], The computational efficiency of Euclid's algorithm has been studied thoroughly. Suppose we wish to compute \(\gcd(27,33)\). There exist 21 quadratic fields in which there Go through the steps and find the GCF of positive integers a, b where a>b. Thus, any other number c that divides both a and b must also divide g. The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c.[4]. For instance, one of the standard proofs of Lagrange's four-square theorem, that every positive integer can be represented as a sum of four squares, is based on quaternion GCDs in this way. The quotients qk are generally found by rounding the real and complex parts of the exact ratio (such as the complex number /) to the nearest integers. MP Board Books in English, Hindi | Madhya Pradesh Board Textbooks for Classes 1 to 12, Tesla Plans To Build Factory in Mexico Worth Over US$5 Billions Versionweekly.com, Buying Textbooks for School? Hence we can find \(\gcd(a,b)\) by doing something that most people learn in The validity of the Euclidean algorithm can be proven by a two-step argument. 154 = (3)41 + 31 154 = ( 3) 41 + 31. For example, 21 is the GCD of 252 and 105 (as 252=2112 and 105=215), and the same number 21 is also the GCD of 105 and 252105=147. [13] The final nonzero remainder is the greatest common divisor of a and b: r Two such multiples can be subtracted (q0=2), leaving a remainder of 147: Then multiples of 147 are subtracted from 462 until the remainder is less than 147. values (Bach and Shallit 1996). Diophantine equations are equations in which the solutions are restricted to integers; they are named after the 3rd-century Alexandrian mathematician Diophantus. Thus every two steps, the numbers algorithms have now been discovered. Let values of x and y calculated by the recursive call be x1 and y1. GCD Calculator - Online Tool (with steps) GCD Calculator: Euclidean Algorithm How to calculate GCD with Euclidean algorithm a a and b b are two integers, with 0 b< a 0 b < a . [3] For example, 6 and 35 factor as 6=23 and 35=57, so they are not prime, but their prime factors are different, so 6 and 35 are coprime, with no common factors other than 1. ax + by = gcd(a, b)gcd(a, b) = gcd(b%a, a)gcd(b%a, a) = (b%a)x1 + ay1ax + by = (b%a)x1 + ay1ax + by = (b [b/a] * a)x1 + ay1ax + by = a(y1 [b/a] * x1) + bx1, Comparing LHS and RHS,x = y1 b/a * x1y = x1. Indeed, if a = a 0d and b = b0d for some integers a0 and b , then ab = (a0 b0)d; hence, d divides . Since the degree is a nonnegative integer, and since it decreases with every step, the Euclidean algorithm concludes in a finite number of steps. Following these instructions I wrote a . The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. The Euclidean algorithm can be used to arrange the set of all positive rational numbers into an infinite binary search tree, called the SternBrocot tree. It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, Bureau 42: [152] Lam's approach required the unique factorization of numbers of the form x + y, where x and y are integers, and = e2i/n is an nth root of 1, that is, n = 1. The version of the Euclidean algorithm described above (and by Euclid) can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. with . are just remainders, so the algorithm can be easily Continued fraction factorization uses continued fractions, which are determined using Euclid's algorithm. Many of the applications described above for integers carry over to polynomials. A useful way to understand the extended Euclidean algorithm is in terms of linear algebra. The Euclidean algorithm developed for two Gaussian integers and is nearly the same as that for ordinary integers,[140] but differs in two respects. Thus the algorithm must eventually produce a zero remainder rN = 0. Thus, they have the form u + v, where u and v are integers and has one of two forms, depending on a parameter D. If D does not equal a multiple of four plus one, then, If, however, D does equal a multiple of four plus one, then. Several other integer relation 12 6 = 2 remainder 0. (If negative inputs are allowed, or if the mod function may return negative values, the last line must be changed into return max(a, a).). To use Euclid's algorithm, divide the smaller number by the larger number. We then attempt to tile the residual rectangle with r0r0 square tiles. [5] Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly. As it turns out (for me), there exists an Extended Euclidean algorithm. Suppose \(x' ,y'\) is another solution. as may be seen by dividing all the steps in the Euclidean algorithm by g.[94] By the same argument, the number of steps remains the same if a and b are multiplied by a common factor w: T(a, b) = T(wa, wb). step we get a remainder \(r' \le b / 2\). Greek mathematician Euclid invented the procedure of repeated application of division to find the GCF or GCD. First, if \(d\) divides \(a\) and \(d\) divides \(b\), then Welcome to MathPortal. Unlike many other calculators out there this provides detailed steps explaining every minute detail. A Therefore, the number of steps T may vary dramatically between neighboring pairs of numbers, such as T(a, b) and T(a,b+1), depending on the size of the two GCDs. [12] For example. GCD Calculator - Greatest Common Divisor (for up to 20 numbers) [158] In other words, there are numbers and such that. There are several methods to find the GCF of a number while some being simple and the rest being complex. + (2*n 1)^2, Sum of the series 0.6, 0.06, 0.006, 0.0006, to n terms, Minimum digits to remove to make a number Perfect Square, Print first k digits of 1/n where n is a positive integer, Check if a given number can be represented in given a no. [154][155] The cases D = 1 and D = 3 yield the Gaussian integers and Eisenstein integers, respectively. [10] The greatest common divisor g of two nonzero numbers a and b is also their smallest positive integral linear combination, that is, the smallest positive number of the form ua+vb where u and v are integers. Nevertheless, these general operations should respect many of the laws governing ordinary arithmetic, such as commutativity, associativity and distributivity. Euclidean algorithm - Wikipedia | The operations are called addition, subtraction, multiplication and division and have their usual properties, such as commutativity, associativity and distributivity. 2260 816 = 2 R 628 (2260 = 2 816 + 628) Since \(x a + y b\) is a multiple of \(d\) for any integers \(x, y\), A set of elements under two binary operations, denoted as addition and multiplication, is called a Euclidean domain if it forms a commutative ring R and, roughly speaking, if a generalized Euclidean algorithm can be performed on them. The Euclidean algorithm has a close relationship with continued fractions. If the ratio of a and b is very large, the quotient is large and many subtractions will be required. can be given as follows. We denote the greatest common divisor of \(a\) and \(b\) by \(\gcd(a,b)\), or In mathematics, the Euclidean algorithm,[note 1] or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. [109], A third average Y(n) is defined as the mean number of steps required when both a and b are chosen randomly (with uniform distribution) from 1 to n[108], Substituting the approximate formula for T(a) into this equation yields an estimate for Y(n)[110], In each step k of the Euclidean algorithm, the quotient qk and remainder rk are computed for a given pair of integers rk2 and rk1, The computational expense per step is associated chiefly with finding qk, since the remainder rk can be calculated quickly from rk2, rk1, and qk, The computational expense of dividing h-bit numbers scales as O(h(+1)), where is the length of the quotient. The GCD of three or more numbers equals the product of the prime factors common to all the numbers,[11] but it can also be calculated by repeatedly taking the GCDs of pairs of numbers. [75] This fact can be used to prove that each positive rational number appears exactly once in this tree. The Gaussian integers are complex numbers of the form = u + vi, where u and v are ordinary integers[note 2] and i is the square root of negative one. By adding/subtracting u multiples of the first cup and v multiples of the second cup, any volume ua+vb can be measured out. Thus, N5log10b. The numbers must be separated by commas, spaces or tabs or may be entered on separate lines. The algorithm In the subtraction-based version, which was Euclid's original version, the remainder calculation (b:=a mod b) is replaced by repeated subtraction. For illustration, the Euclidean algorithm can be used to find the greatest common divisor of a=1071 and b=462. This calculator uses Euclid's Algorithm to determine the multiple. with the two numbers of interest (with the larger of the two written first). [135], For example, consider the following two quartic polynomials, which each factor into two quadratic polynomials. evaluates to. r To do this, we choose the largest integer first, i.e. We reconsider example 2 above: N = 195 and P = 154. The calculator gives the greatest common divisor (GCD) of two input polynomials. Step 1: On dividing 78 66 you will have the quotient 1 and remainder 12. Step 4: The GCD of 84 and 140 is: Search our database of more than 200 calculators. Therefore, a=q0b+r0b+r0FM+1+FM=FM+2, The GCD calculator allows you to quickly find the greatest common divisor of a set of numbers. The unique factorization of Euclidean domains is useful in many applications. The Euclidean Algorithm. divide \(a\) by \(b\) to get \(a = b q + r\), and \(r > b / 2\), then in the next x and y are updated using the below expressions. If you want to contact me, probably have some questions, write me using the contact form or email me on The result is a continued fraction, In the worked example above, the gcd(1071, 462) was calculated, and the quotients qk were 2, 3 and 7, respectively. If you're used to a different notation, the output of the calculator might confuse you at first. ) for integers \(x\) and \(y\)? Enter the numbers you want to find the GCF or HCF and click on the Calculate Button to get the result in a short span of time. because it divides both terms on the right-hand side of the equation. [136] The Euclidean algorithm can be used to solve linear Diophantine equations and Chinese remainder problems for polynomials; continued fractions of polynomials can also be defined. [62] Specifically, if a prime number divides L, then it must divide at least one factor of L. Conversely, if a number w is coprime to each of a series of numbers a1, a2, , an, then w is also coprime to their product, a1a2an. [98] For if the algorithm requires N steps, then b is greater than or equal to FN+1 which in turn is greater than or equal to N1, where is the golden ratio. The common divisors can be found by dividing both numbers by successive integers from 2 to the smaller number b. 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As before, we set r2 = and r1 = , and the task at each step k is to identify a quotient qk and a remainder rk such that, where every remainder is strictly smaller than its predecessor: |rk| < |rk1|. [28] The algorithm was probably known by Eudoxus of Cnidus (about 375 BC). The recursive nature of the Euclidean algorithm gives another equation, If the Euclidean algorithm requires N steps for a pair of natural numbers a>b>0, the smallest values of a and b for which this is true are the Fibonacci numbers FN+2 and FN+1, respectively. Euclid's algorithm calculates the greatest common divisor of two positive integers a and b. {\displaystyle \left|{\frac {r_{k+1}}{r_{k}}}\right|<{\frac {1}{\varphi }}\sim 0.618,} We will show them using few examples. {\displaystyle \varphi } If r is not equal to zero then apply Euclids Division Lemma to b and r. Step 3: Continue the Process until the remainder is zero. [88][89], In the uniform cost model (suitable for analyzing the complexity of gcd calculation on numbers that fit into a single machine word), each step of the algorithm takes constant time, and Lam's analysis implies that the total running time is also O(h). have been substituted, the final equation expresses g as a linear sum of a and b, so that g=sa+tb. [26][27] The mathematician and historian B. L. van der Waerden suggests that Book VII derives from a textbook on number theory written by mathematicians in the school of Pythagoras. [40] Gauss mentioned the algorithm in his Disquisitiones Arithmeticae (published 1801), but only as a method for continued fractions. The Euclidean algorithm is a way to find the greatest common divisor of two positive integers. gcd Everyone who receives the link will be able to view this calculation, Copyright PlanetCalc Version: Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers. r The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. In particular, the computation of the modular multiplicative inverse is an essential step in RSA public-key encryption method. Certain problems can be solved using this result. 2, 3, are 1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 4, 5, (OEIS A034883). 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