"period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers. The smallest pair of amicable numbers is (220, 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084),"
"9 9 (nine) is the natural number following and preceding . 9 is a composite number, its proper divisors being and . It is 3 times 3 and hence the third square number. Nine is a Motzkin number. It is the first composite lucky number, along with the first composite odd number and only single-digit composite odd number. 9 is the only positive perfect power that is one more than another positive perfect power, by Mihăilescu's Theorem. 9 is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (""p"") and the"
"911 (number) 911 (nine hundred [and] eleven) is the integer following 910 and preceding 912. It is a prime number, a Sophie Germain prime and the sum of three consecutive primes (293 + 307 + 311). It is an Eisenstein prime with no imaginary part and real part of the form 3""n"" − 1. Since 913 is a semiprime, 911 is a Chen prime. It is also a centered decagonal number. There are 911 inverse semigroups of order 7 911 is obtained by concatenating its product of digits and sum of digits. As ""911"" or ""9/11"", typically pronounced ""nine-eleven"", it"
"10), (6, 12), (8, 10), (8, 12), and (10, 12). Since the sums ""u"" + ""v"" and ""v"" + ""u*"" are even, the sums ""a"" + ""b"" + ""c"" and ""d"" + ""e"" + ""f"" should be even as well. The only way that the sum of three integers will result in an even number is when 1) two of them are odd and one is even, or 2) when all three are even. The fact that the two corner cells are even means that we have only 2 even numbers at our disposal. Thus, the second statement is not"
"0.999... In mathematics, 0.999... (also written 0., among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...). This number is equal to 1. In other words, ""0.999..."" and ""1"" represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on target audience, background assumptions, historical context, and preferred development of the real numbers, the system within"
"often in Indian culture and mythology. Some instances are enumerated below. A human pregnancy normally lasts nine months, the basis of Naegele's rule. 9 9 (nine) is the natural number following and preceding . 9 is a composite number, its proper divisors being and . It is 3 times 3 and hence the third square number. Nine is a Motzkin number. It is the first composite lucky number, along with the first composite odd number and only single-digit composite odd number. 9 is the only positive perfect power that is one more than another positive perfect power, by Mihăilescu's Theorem."
"Germain prime, the sum of seven consecutive primes (71 + 73 + 79 + 83 + 89 + 97 + 101), the sum of nine consecutive primes (47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83), and an Eisenstein prime with no imaginary part. 593 is also notable for being the sum of 9 + 2 (thus a Leyland number). 593 (number) 593 (five hundred [and] ninety-three) is the natural number following 592 and preceding 594. 593 is an odd number. It is a prime number, an example of what Paul Erdős"
"a single digit. The digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value. Digit sums and digital roots can be used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum (or digital root) is divisible by 3 or 9, respectively. For divisibility by 9, this test is called the rule of nines and is the basis of the casting out nines technique for checking calculations. Digit sums are also a common ingredient in checksum"
"convergence is repeated in many textbooks, for example Hardy and Wright and Apostol. We group the terms of the sum by the number of digits in the denominator. The number of ""n""-digit positive integers that have no digit equal to '9' is 8(9) because there are 8 choices (1 through 8) for the first digit, and 9 independent choices (0 through 8) for each of the other ""n""−1 digits. Each of these numbers having no '9' is greater than or equal to 10, so the reciprocal of each of these numbers is less than or equal to 10. Therefore, the"
"prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even. Intersection is denoted by the from Unicode Mathematical Operators. Intersection In mathematics, the intersection of two or more objects is another, usually ""smaller"" object. All objects are presumed to lie in a certain common space except in set theory, where the intersection of arbitrary sets is defined. The intersection is one of basic concepts of geometry. Intuitively, the intersection of two or more objects is a new object that lies in each of"
"first that is odd. All subsequent squares of this form are odd. Since , 9 is an exponential factorial. A polygon with nine sides is called a nonagon or enneagon. A group of nine of anything is called an ennead. In base 10 a positive number is divisible by 9 if and only if its digital root is 9. That is, if any natural number is multiplied by 9, and the digits of the answer are repeatedly added until it is just one digit, the sum will be nine: There are other interesting patterns involving multiples of nine: This works"
"It began on October 25, 1993. The current format utilizes three sets of digits 0 through 9. Players choose: ""exact order"" (all three digits), ""any order"" (two or three digits), ""exact and any order"" (two or three digits), ""combo"" (two or three digits; ""combo two"" is the same as two exact order plays while ""combo"" three is the same as ""three or six"" exact order plays) and ""Sum It Up"" (the arithmetic sum of the numbers played is the same as the arithmetic sum of the numbers drawn; the numbers drawn do not have to be the same as the"
"are infinitely many Mersenne primes. As well as having the form 2(2 − 1), each even perfect number is the triangular number (and hence equal to the sum of the integers from 1 to ) and the hexagonal number. Furthermore, each even perfect number except for 6 is the centered nonagonal number and is equal to the sum of the first odd cubes: Even perfect numbers (except 6) are of the form with each resulting triangular number T = 28, T = 496, T = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending"
"4,294,967,295 The number 4,294,967,295 is an integer equal to 2 − 1. It is a perfect totient number. It follows 4,294,967,294 and precedes 4,294,967,296. It has a factorization of formula_1. Since the prime factors of 2 − 1 are exactly the five known Fermat primes, this number is the largest known odd value ""n"" for which a regular ""n""-sided polygon is constructible using compass and straightedge. Equivalently, it is the largest known odd number ""n"" for which the angle formula_2 can be constructed, or for which formula_3 can be expressed in terms of square roots. Not only is 4,294,967,295 the"
"a number between 1 and 10 using a two-out-of-five code with the following numeric equivalence: 1, 2, 4, 7, 0. The numbers are summed together. For example, the number 6 is encoded NWWNN, with wide bars occupying the positions for 2 and 4 (2+4=6). In the case of NNWWN which is it is assigned to 0 for digits (+0), and 10 for the letter columns (+10 – +30). When encoding the (+10 to +30) Letters the equation needs a ""−1"" added so 'A' is WNNNW → 1 + 10 − 1 → 10 as shown in the table. The last"
"29 (number) 29 (twenty-nine) is the natural number following 28 and preceding 30. It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three consecutive squares, 2 + 3 + 4. It is a Lucas prime, a Pell prime, and a tetranacci number. It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 29 is also the 10th supersingular prime. None of"
"A − C ≤ 9, both the first and third digits are guaranteed to be single digits.) Its reversal is 100 × ['] + 10 × ' + 1 × [']. The sum is thus 101 × ['] + 20 × ' + 101 × ['] = 101 × [' + '] + 20 × = 101 × [−1 + 10] + 180 = 1089. Multiplying the number 1089 by the integers from 1 to 9 produces a pattern: multipliers adding up to 10 give products that are the digit reversals of each other: Also note the patterns within each"
"9: This system yields this maximum: 8 × 9² + 8 × 9 + 8 × 9° = 648 + 72 + 8 = 728 = 9³ − 1 Examples: 3 throws, random selection of one of the dice for each throw, type is used as attribute: 5832 = 2 × 2 × 2 × 9 × 9 × 9 = 18³ numbers are possible. Since the middle of the 1980s, the press wrote about the games. Winkelmann presented games himself, for example, in 1987 in Vienna, at the ""Österrechischen Spielefest, Stiftung Spielen in Österreich"", Leopoldsdorf, where ""Miwin's dice"" won"
"large even number can be written as the sum of a prime and a semiprime (the product of two primes) – e.g., 100 = 23 + 7·11. The Asteroid 7681 Chenjingrun was named after him. In 1999, China issued an 80-cent postage stamp, titled ""The Best Result of Goldbach Conjecture,"" with a silhouette of Chen and the inequality: Several statues in China have been built in memory of Chen. At Xiamen University, the names of Chen and four other mathematicians — Peter Gustav Lejeune Dirichlet, Matti Jutila, Yuri Linnik, and Pan Chengdong — are inscribed in the marble slab behind"
"graph ""K"" has 229 crossings in its straight-line drawing with the fewest possible crossings. 229 (number) 229 (two hundred [and] twenty-nine) is the natural number following 228 and preceding 230. It is a prime number, and a regular prime. It is also a full reptend prime, meaning that the decimal expansion of the unit fraction 1/229 repeats periodically with as long a period as possible. With 227 it is the larger of a pair of twin primes, and it is also the start of a sequence of three consecutive squarefree numbers. It is the smallest prime that, when added to"
"159th congruent number. It is the sum of the 52nd prime number, 239, and 52, making it part of the formula_1 series. 291 was the name of a New York City-based arts and literature magazine from 1915 to 1916, the name of which was inspired off the name of an art gallery owned by its publisher. Both names come from the gallery's address at 291 Fifth Avenue. 291 Alice is a main belt asteroid discovered in 1890. See also the year 291. 291 (number) 291 (two hundred [and] ninety one) is the natural number following 290 and preceding 292. This"
"multiplication of even and odd integers obey simple rules. The addition or subtraction of two even numbers or of two odd numbers always produces an even number, e.g., 4 + 6 = 10 and 3 + 5 = 8. Conversely, the addition or subtraction of an odd and even number is always odd, e.g., 3 + 8 = 11. The multiplication of two odd numbers is always odd, but the multiplication of an even number with any number is always even. An odd number raised to a power is always odd and an even number raised to power is always"
"9 and must be such that the sum of all the thirteen digits, each multiplied by its (integer) weight, alternating between 1 and 3, is a multiple of 10. Formally, using modular arithmetic, we can say: The calculation of an <nowiki>ISBN-13</nowiki> check digit begins with the first 12 digits of the thirteen-digit ISBN (thus excluding the check digit itself). Each digit, from left to right, is alternately multiplied by 1 or 3, then those products are summed modulo 10 to give a value ranging from 0 to 9. Subtracted from 10, that leaves a result from 1 to 10. A"
"for all the multiples of 9. is the only other such that a number is divisible by ""n"" if and only if its digital root is divisible by ""n"". In base-""N"", the divisors of have this property. Another consequence of 9 being , is that it is also a Kaprekar number. The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples: Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers, known as long ago as the 12th century. Six recurring"
"a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc. Unlike a regular based numeral system, there are numbers like 9b where 9 and b each represents 35; yet the representation is unique because ac and aca are not allowed – the a would terminate the number. The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes. The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of"
"may be interpreted as 10 in this situation. A fairly consistent arrangement of the faces on ten-digit dice has been observed. If one holds such a die between one's fingers at two of the vertices such that the even numbers are on top, and reads the numbers from left to right in a zigzag pattern, the sequence obtained is 0, 7, 4, 1, 6, 9, 2, 5, 8, 3, and back to 0. The even and odd digits are divided among the two opposing ""caps"" of the die, and each pair of opposite faces adds to nine. When casting a"
"is commonly used to refer the calendar dates November 9 or September 11, depending on which date notation is used. The latter usage most commonly refers to the North American emergency telephone number, see 9-1-1, or the September 11, 2001 attacks in the United States. 911 (number) 911 (nine hundred [and] eleven) is the integer following 910 and preceding 912. It is a prime number, a Sophie Germain prime and the sum of three consecutive primes (293 + 307 + 311). It is an Eisenstein prime with no imaginary part and real part of the form 3""n"" − 1. Since"
"229 (number) 229 (two hundred [and] twenty-nine) is the natural number following 228 and preceding 230. It is a prime number, and a regular prime. It is also a full reptend prime, meaning that the decimal expansion of the unit fraction 1/229 repeats periodically with as long a period as possible. With 227 it is the larger of a pair of twin primes, and it is also the start of a sequence of three consecutive squarefree numbers. It is the smallest prime that, when added to the reverse of its decimal representation, yields another prime: 229 + 922 = 1151."
"the spectator choose any two suitable numbers and then apply some basic math to produce a single four-digit number. That number is always 1089. The spectator is then asked to turn to page 108 of a book and read the 9th word, which the magician has memorized. To the audience it looks like the number is random, but through manipulation, the result is always the same. It is this property that led University of Oxford mathematician David Acheson to title his 2010 book '1089 and all that: a journey into mathematics'. In base 10, the following steps always yield 1089:"
"total number of fingers is zero, so the even player wins. One teachers' manual suggests playing this game as a way to introduce children to the concept that 0 is divisible by 2. Parity of zero Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of ""even"": it is an integer multiple of 2, specifically . As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal"
"difference between 9 and any higher odd perfect square is eight times a triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of differ by an amount containing an odd factor, the only perfect square of the form is 1, and the only perfect square of the form is 9. Square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be"
"that does not begin with 1 or 0, simply add a constant value to bring it into that range (to obtain random numbers between 8 and 16, inclusive, follow the 1 – 9 instructions below, and add 7 to the result of each roll). Governing probability: P(0) = P(1) = ... = P(90) = 8/9³ = 8/729 To obtain an equal distribution with numbers from 0 – 90, all three dice are rolled, one at a time, in a random order. The result is calculated based on the following rules: Sample: This gives 91 numbers, from 0 – 90 with"
"Parity (mathematics) In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is evenly divisible by two and odd if it is not even. For example, 6 is even because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of even numbers include −4, 0, 82 and 178. In particular, zero is an even number. Some examples of odd numbers are −5, 3, 29, and 73. A formal definition of"