Question:

What can 147 and 64 go into?

Answer:

The Least Common Multiplier (LCM) of the numbers 147 and 64 is 9408. The number 1 is their Greatest Common Factor (GCF).

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Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder. The name "divisor" comes from the arithmetic operation of division: if \frac{a}{b} = c then a is the dividend, b the divisor, and c the quotient. In general, for non-zero integers m and n, it is said that m divides n—and, dually, that n is divisible by m—written: if there exists an integer k such that n = km. Thus, divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.) 1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also division by zero). Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd. 1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non-trivial divisor. A number with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules which allow one to recognize certain divisors of a number from the number's digits. The generalization can be said to be the concept of divisibility in any integral domain. There are some elementary rules: If a \mid bc, and gcd(a, b) = 1, then a \mid c. This is called Euclid's lemma. If p is a prime number and p \mid ab then p \mid a or p \mid b. A positive divisor of n which is different from n is called a proper divisor or an aliquot part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n. An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself. Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of the fundamental theorem of arithmetic. A number n is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than n, and abundant if this sum exceeds n. The total number of positive divisors of n is a multiplicative function d(n), meaning that when two numbers m and n are relatively prime, then d(mn)=d(m)\times d(n). For instance, d(42) = 8 = 2 \times 2 \times 2 = d(2) \times d(3) \times d(7); the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicative function: if the two numbers m and n share a common divisor, then it might not be true that d(mn)=d(m)\times d(n). The sum of the positive divisors of n is another multiplicative function \sigma (n) (e.g. \sigma (42) = 96 = 3 \times 4 \times 8 = \sigma (2) \times \sigma (3) \times \sigma (7) = 1+2+3+6+7+14+21+42). Both of these functions are examples of divisor functions. If the prime factorization of n is given by then the number of positive divisors of n is and each of the divisors has the form where  0 \le \mu_i \le \nu_i for each 1 \le i \le k. For every natural n, d(n) < 2 \sqrt{n}. Also, where  \gamma is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about \ln n. The relation of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.

Least common multiple
In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM(ab) is defined to be zero. The LCM is familiar from grade-school arithmetic as the "least common denominator" (LCD) that must be determined before fractions can be added, subtracted or compared. The LCM of more than two integers is also well-defined: it is the smallest integer that is divisible by each of them. A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and 2 as well. In this article we will denote the least common multiple of two integers a and b as lcm( a, b ). Some older textbooks use [ a, b ]. What is the LCM of 4 and 6? Multiples of 4 are: and the multiples of 6 are: Common multiples of 4 and 6 are simply the numbers that are in both lists: So, from this list of the first few common multiples of the numbers 4 and 6, their least common multiple is 12. When adding, subtracting, or comparing vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator, because each of the fractions can be expressed as a fraction with this denominator. For instance, where the denominator 42 was used because it is the least common multiple of 21 and 6. Many school age children are taught the term greatest common factor (GCF) instead of the greatest common divisor(GCD); therefore, for those familiar with the concept of GCF, substitute GCF when GCD is used below. The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (GCD): This formula is also valid when exactly one of a and b is 0, since gcd(a, 0) = |a|. There are fast algorithms for computing the GCD that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above, Because gcd(a, b) is a divisor of both a and b, it's more efficient to compute the LCM by dividing before multiplying: This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results (overflow in the a×b computation). Because gcd(a, b) is a divisor of both a and b, the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer. Done this way, the previous example becomes: The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number. For example: Here we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5. This knowledge can be used to find the LCM of a set of numbers. Example: Find the value of lcm(8,9,21). First, factor out each number and express it as a product of prime number powers. The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 23, 32, and 71, respectively. Thus, This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization, but is useful for illustrating concepts. This method can be illustrated using a Venn diagram as follows. Find the prime factorization of each of the two numbers. Put the prime factors into a Venn diagram with one circle for each of the two numbers, and all factors they share in common in the intersection. To find the LCM, just multiply all of the prime numbers in the diagram. Here is an example: and what they share in common is two "2"s and a "3": This also works for the greatest common divisor (GCD), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the GCD of 48 and 180 is 2 × 2 × 3 = 12. This method works as easily for finding the LCM of several integers. Let there be a finite sequence of positive integers X = (x1, x2, ..., xn), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X)m( = (x1)m(, x2)m(, ..., xn)m(), X(1) = X, where X)m( is the mth iteration of X, i.e. X at step m of the algorithm, etc. The purpose of the examination is to pick up the least (perhaps, one of many) element of the sequence X)m(. Assuming xk0)m( is the selected element, the sequence X+1)m( is defined as In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X)m( to X+1)m( unchanged. The algorithm stops when all elements in sequence X)m( are equal. Their common value L is exactly LCM(X). (For a proof and an interactive simulation see reference below, Algorithm for Computing the LCM.) This method works for any number of factors. One begins by listing all of the numbers vertically in a table (in this example 4, 7, 12, 21, and 42): The process begins by dividing all of the factors by 2. If any of them divides evenly, write 2 at the top of the table and the result of division by 2 of each factor in the space to the right of each factor and below the 2. If a number does not divide evenly, just rewrite the number again. If 2 does not divide evenly into any of the numbers, try 3. Now, check if 2 divides again: Once 2 no longer divides, divide by 3. If 3 no longer divides, try 5 and 7. Keep going until all of the numbers have been reduced to 1. Now, multiply the numbers on the top and you have the LCM. In this case, it is 2 × 2 × 3 × 7 = 84. You will get to the LCM the quickest if you use prime numbers and start from the lowest prime, 2. According to the fundamental theorem of arithmetic a positive integer is the product of prime numbers, and, except for their order, this representation is unique: where the exponents n2, n3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ... Given two integers   \;a=\prod_p p^{a_p}\;\;   and   \;b=\prod_p p^{b_p}\;\;   their least common multiple and greatest common divisor are given by the formulas and Since this gives In fact, any rational number can be written uniquely as the product of primes if negative exponents are allowed. When this is done, the above formulas remain valid. Using the same examples as above: The positive integers may be partially ordered by divisibility: if a divides b (i.e. if b is an integer multiple of a) write ab (or equivalently, ba). (Forget the usual magnitude-based definition of ≤ in this section - it isn't used.) Under this ordering, the positive integers become a lattice with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them: The following pairs of dual formulas are special cases of general lattice-theoretic identities. It can also be shown that this lattice is distributive, i.e. that lcm distributes over gcd and, dually, that gcd distributes over lcm: This identity is self-dual: Let D be the product of ω(D) distinct prime numbers (i.e. D is squarefree). Then where the absolute bars || denote the cardinality of a set. The least common multiple can be defined generally over commutative rings as follows: Let a and b be elements of a commutative ring R. A common multiple of a and b is an element m of R such that both a and b divide m (i.e. there exist elements x and y of R such that ax = m and by = m). A least common multiple of a and b is a common multiple that is minimal in the sense that for any other common multiple n of a and b, m divides n. In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b (the intersection of a collection of ideals is always an ideal). In principal ideal domains, one can even talk about the least common multiple of arbitrary collections of elements: it is a generator of the intersection of the ideals generated by the elements of the collection.

Greatest common divisor
In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more integers (at least one of which is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4. This notion can be extended to polynomials, see Polynomial greatest common divisor, or to rational numbers (with integer quotients). In this article we will denote the greatest common divisor of two integers a and b as gcd(a,b). Some older textbooks use (a,b). The number 54 can be expressed as a product of two other integers in several different ways: Thus the divisors of 54 are: Similarly the divisors of 24 are: The numbers that these two lists share in common are the common divisors of 54 and 24: The greatest of these is 6. That is the greatest common divisor of 54 and 24. One writes: The greatest common divisor is useful for reducing fractions to be in lowest terms. For example, gcd(42, 56) = 14, therefore, Two numbers are called relatively prime, or coprime if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime. For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, with two squares along one edge (24/12 = 2) and five squares along the other (60/12 = 5). Greatest common divisors can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example: to compute gcd(18, 84), we find the prime factorizations 18 = 2 · 32 and 84 = 22 · 3 · 7 and notice that the "overlap" of the two expressions is 2 · 3; so gcd(18, 84) = 6. In practice, this method is only feasible for small numbers; computing prime factorizations in general takes far too long. Here is another concrete example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180. First, find the prime factorizations of the two numbers: What they share in common is two "2"s and a "3": A much more efficient method is the Euclidean algorithm, which uses a division algorithm such as long division in combination with the observation that the gcd of two numbers also divides their difference. To compute gcd(48,18), divide 48 by 18 to get a quotient of 2 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd. Note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as: where If the arguments are both greater than zero then the algorithm can be written in more elementary terms as follows: The existence of the Euclidean algorithm places (the decision problem version of) the greatest common divisor problem in P, the class of problems solvable in polynomial time. The GCD problem is not known to be in NC, and so there is no known way to parallelize its computation across many processors; nor is it known to be P-complete, which would imply that it is unlikely to be possible to parallelize GCD computation. In this sense the GCD problem is analogous to e.g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete. Shallcross et al. showed that a related problem (EUGCD, determining the remainder sequence arising during the Euclidean algorithm) is NC-equivalent to the problem of integer linear programming with two variables; if either problem is in NC or is P-complete, the other is as well. Since NC contains NL, it is also unknown whether a space-efficient algorithm for computing the GCD exists, even for nondeterministic Turing machines. Although the problem is not known to be in NC, parallel algorithms with time superior to the Euclidean algorithm exist; the best known deterministic algorithm is by Chor and Goldreich, which (in the CRCW-PRAM model) can solve the problem in O(n/log n) time with n1+ε processors. Randomized algorithms can solve the problem in O((log n)2) time on \exp\left[O\left(\sqrt{n \log n}\right)\right] processors (note this is superpolynomial). An alternative method of computing the gcd is the binary gcd method which uses only subtraction and division by 2. In outline the method is as follows: Let a and b be the two non negative integers and (without loss of generality) let ab. There are now four possibilities In this case 2 is a common factor. Divide both a and b by 2 and continue. In this case 2 is not a common factor. Divide a by 2 and continue. Like the previous case 2 is not a common factor. Divide b by 2 and continue. In this case let c = ( a - b ) / 2. Then gcd(a,b) = gcd(a,c) = gcd(b,c). Because ba it is usually easier (and computationally faster) to determine the gcd(b,c). If computing this algorithm by hand the gcd(b,c) may be apparent. Otherwise continue the algorithm until either b = c or c = 0. For further details see Binary GCD algorithm. If a and b are not both zero, the greatest common divisor of a and b can be computed by using least common multiple (lcm) of a and b: but more commonly the lcm is computed from the gcd. Using Thomae's function f, which generalizes to a and b rational or commensurate reals. Keith Slavin has shown that for odd a ≥ 1: which is a function that can be evaluated for complex b. Wolfgang Schramm has shown that is an entire function in the variable b for all positive integers a where cd(k) is Ramanujan's sum. Donald Knuth proved the following reduction: for non-negative integers a and b, where a and b are not both zero. More generally which can be proven by considering the Euclidean algorithm in base n. Another useful identity relates \gcd(a,b) to the Euler's totient function: In 1972, James E. Nymann showed that k integers, chosen independently and uniformly from {1,...,n}, are coprime with probability 1/ζ(k) as n goes to infinity. (See coprime for a derivation.) This result was extended in 1987 to show that the probability that k random integers has greatest common divisor d is d-k/ζ(k). Using this information, the expected value of the greatest common divisor function can be seen (informally) to not exist when k = 2. In this case the probability that the gcd equals d is d−2/ζ(2), and since ζ(2) = π2/6 we have This last summation is the harmonic series, which diverges. However, when k ≥ 3, the expected value is well-defined, and by the above argument, it is For k = 3, this is approximately equal to 1.3684. For k = 4, it is approximately 1.1106. The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements. If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b (that is, if there are elements x and y in R such that d·x = a and d·y = b). If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b. Note that with this definition, two elements a and b may very well have several greatest common divisors, or none at all. If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other; indeed if a gcd exists, any one of its associates is a gcd as well. Existence of a gcd is not assured in arbitrary integral domains. However if R is a unique factorization domain, then any two elements have a gcd, and more generally this is true in gcd domains. If R is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance when R = F[X] where F is a field, or when R is the ring of Gaussian integers), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure. The following is an example of an integral domain with two elements that do not have a gcd: The elements 2 and 1 + √(−3) are two "maximal common divisors" (i.e. any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √(−3)), but they are not associated, so there is no greatest common divisor of a and b. Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form pa + qb, where p and q range over the ring. This is the ideal generated by a and b, and is denoted simply (ab). In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element d; then this d is a greatest common divisor of a and b. But the ideal (ab) can be useful even when there is no greatest common divisor of a and b. (Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.)

Supernatural numbers
In mathematics, the supernatural numbers (sometimes called generalized natural numbers or Steinitz numbers) are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a part of his work on field theory. A supernatural number \omega is a formal product: where p runs over all prime numbers, and each n_p is zero, a natural number or infinity. Sometimes we write v_p(\omega) for n_p. If no n_p = \infty and there are only a finite number of non-zero n_p then we recover the natural numbers. Slightly less intuitively, if all n_p are \infty, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide \omega "infinitely often," by taking that prime's corresponding exponent to be the symbol \infty. There is no natural way to add supernatural numbers, but they can be multiplied, with \prod_p p^{n_p}\cdot\prod_p p^{m_p}=\prod_p p^{n_p+m_p}. Similarly, the notion of divisibility extends to the supernaturals with \omega_1\mid\omega_2 if v_p(\omega_1)\leq v_p(\omega_2) for all p. We can also generalize the notion of the least common multiple and greatest common divisor for supernatural numbers, by defining With these definitions, we can now take the gcd or lcm of infinitely many natural numbers to get a supernatural number. We can also extend the usual -adicp order functions to supernatural numbers by defining v_p(\omega)=n_p for each p Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are also used implicitly in many number-theoretical proofs, such as the density of the square-free integers and bounds for odd perfect numbers.

Extravagant number
An extravagant number (also known as a wasteful number) is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents). For example, in base-10 arithmetic 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers. Extravagant numbers can be defined in any base. There are infinitely many extravagant numbers, no matter what base is used.

Remainder
In arithmetic, the remainder (or residue) is the amount "left over" after performing the division of two integers which do not divide evenly, that is, where the result of the division cannot be expressed as an integer. If a and d are natural numbers, with d non-zero, it can be proven that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < d. The number q is called the quotient, while r is called the remainder. See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder. If a and d are integers, with d non-zero, then a remainder is an integer r such that a = qd + r for some integer q, and with |r| < |d|. When defined this way, there are two possible remainders. For example, the division of −42 by −5 can be expressed as either as is usual for mathematicians,][ or So the remainder is then either 3 or −2. This ambiguity in the value of the remainder can be quite serious computationally; for mission critical computing systems, the wrong choice can lead to dangerous consequences. In the case above, the negative remainder is obtained from the positive one just by subtracting 5, which is d. This holds in general. When dividing by d, if the positive remainder is r1, and the negative one is r2, then When a and d are floating-point numbers, with d non-zero, a can be divided by d without remainder, with the quotient being another floating-point number. If the quotient is constrained to being an integer, however, the concept of remainder is still necessary. It can be proved that there exists a unique integer quotient q and a unique floating-point remainder r such that a = qd + r with 0 ≤ r < |d|. As in the case of division of integers, the remainder could be required to be negative, that is, −|d| < r ≤ 0. Extending the definition of remainder for floating-point numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition—see modulo operation. The way remainder was defined, in addition to the equality a = qd + r an inequality was also imposed, which was either 0 ≤ r < |d| or −|d| < r ≤ 0. Such an inequality is necessary in order for the remainder to be unique—that is, for it to be well-defined. The choice of such an inequality is somewhat arbitrary. Any condition of the form x < r ≤ x + |d| (or x ≤ r < x + |d|), where x is a constant, is enough to guarantee the uniqueness of the remainder. With two choices for the inequality, there are two choices for the remainder, one negative and the other positive. This means that there are also two possible choices for the quotient. In number theory the positive remainder is chosen by convention. But programming languages need not, and different languages have adopted different conventions: Pascal goes with number theory in choosing the result of the mod operation positive (while not always a = (a div d )*d + a mod d ). C99 chooses the remainder with the same sign as the dividend a. (Before C99, the C language allowed either choice.) Perl, Python (only modern versions), and Common Lisp choose the remainder with the same sign as the divisor d. Haskell and Scheme offer two functions, remainder and modulo – PL/I has mod and rem, while Fortran has mod and modulo; in each case, the former agrees in sign with the dividend, and the latter with the divisor.

Table of divisors
The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, say, for which n/m is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/3 = 7 (and 7 is also a divisor of 21). If m is a divisor of n then so is −m. The tables below only list positive divisors.

Least common multiple

In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility.

The LCM is familiar from grade-school arithmetic as the "least common denominator" (LCD) that must be determined before fractions can be added, subtracted or compared.


Greatest common divisor

In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more integers (at least one of which is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.

This notion can be extended to polynomials, see Polynomial greatest common divisor, or to rational numbers (with integer quotients).

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