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Divisibility Rule
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in ''Scientific American''. Divisibility rules for numbers 1–30 The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last ''n'' digits) the result must be examined by other means. For divisors with multiple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9 (number)
9 (nine) is the natural number following and preceding . Evolution of the Arabic digit In the beginning, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a -look-alike. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ encircles a lowercase ''a''. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic. While the shape of the glyph for the digit 9 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in . The mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
22 (number)
22 (twenty-two) is the natural number following 21 and preceding 23. In mathematics 22 is a palindromic number and the eighth semiprime; its proper divisors are 1, 2, and 11. It is the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number. It is also a Perrin number, from a sum of 10 and 12. 22 is the fourth pentagonal number, the third hexagonal pyramidal number, and the third centered heptagonal number. The maximum number of regions into which five intersecting circles divide the plane is 22. 22 is also the quantity of pieces in a disc that can be created with six straight cuts, which makes 22 the seventh central polygonal number. \frac is a commonly used approximation of the irrational number , the ratio of the circumference of a circle to its diameter; where both 22 and 7 are consecutive hexagonal pyramidal numbers. 22 also features in another approximation for pi, here by Srinivasa Ramanujan from an approximate constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21 (number)
21 (twenty-one) is the natural number following 20 and preceding 22. The current century is the 21st century AD, under the Gregorian calendar. In mathematics 21 is: * a composite number, its proper divisors being 1, 3 and 7, and a deficient number as the sum of these divisors is less than the number itself. * a Fibonacci number as it is the sum of the preceding terms in the sequence, 8 and 13. * the fifth Motzkin number. * a triangular number, because it is the sum of the first six natural numbers (1 + 2 + 3 + 4 + 5 + 6 = 21). * an octagonal number. * a Padovan number, preceded by the terms 9, 12, 16 (it is the sum of the first two of these) in the padovan sequence. * a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes. * the sum of the divisors of the first 5 positive integers (i.e., 1 + (1 + 2) + (1 + 3) + (1 + 2 + 4) + (1 + 5)) * the smallest non-trivial example of a Fibonacci number whose digits are Fibonacci numbers and whose digit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
20 (number)
20 (twenty; Roman numeral XX) is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score. In mathematics *20 is a pronic number. *20 is a tetrahedral number as 1, 4, 10, 20. *20 is the basis for vigesimal number systems. *20 is the third composite number to be the product of a squared prime and a prime, and also the second member of the (''2''2)''q'' family in this form. *20 is the smallest primitive abundant number. *An icosahedron has 20 faces. A dodecahedron has 20 vertices. *20 can be written as the sum of three Fibonacci numbers uniquely, i.e. 20 = 13 + 5 + 2. *20 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. (e.g. the newspaper headline "Scores of Typhoon Survivors Flown to Manila")."CBS News"''Scores of Typhoon Survivors Flown to Manila'' (November 2013) In sports * Twenty20 is a form of limited overs cricket where each team plays only 20 overs. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19 (number)
19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number. Mathematics 19 is the eighth prime number, and forms a sexy prime with 13, a twin prime with 17, and a cousin prime with 23. It is the third full reptend prime, the fifth central trinomial coefficient, and the seventh Mersenne prime exponent. It is also the second Keith number, and more specifically the first Keith prime. * 19 is the maximum number of fourth powers needed to sum up to any natural number, and in the context of Waring's problem, 19 is the fourth value of g(k). * The sum of the squares of the first 19 primes is divisible by 19. *19 is the sixth Heegner number. 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of nine total. * 19 is the third centered triangular number as well as the third centered hexagonal number. : The 19th triangular number is 190, equivalently the sum of the first 19 non-zero integers, that is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18 (number)
18 (eighteen) is the natural number following 17 and preceding 19. In mathematics * Eighteen is a composite number, its divisors being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a semiperfect number. Eighteen is the first inverted square-prime of the form ''p''·''q''2. * In base ten, it is a Harshad number. * It is an abundant number, as the sum of its proper divisors is greater than itself (1+2+3+6+9 = 21). It is known to be a solitary number, despite not being coprime to this sum. * It is the number of one-sided pentominoes. * It is the only number where the sum of its written digits in base 10 (1+8 = 9) is equal to half of itself (18/2 = 9). * It is a Fine number. In science Chemistry * Eighteen is the atomic number of argon. * Group 18 of the periodic table is called the noble gases. * The 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of metal complexes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
17 (number)
17 (seventeen) is the natural number following 16 (number), 16 and preceding 18 (number), 18. It is a prime number. Seventeen is the sum of the first four prime numbers. In mathematics 17 is the seventh prime number, which makes seventeen the fourth super-prime, as seven is itself prime. The next prime is 19 (number), 19, with which it forms a twin prime. It is a cousin prime with 13 (number), 13 and a sexy prime with 11 (number), 11 and 23 (number), 23. It is an emirp, and more specifically a permutable prime with 71 (number), 71, both of which are also supersingular prime (moonshine theory), supersingular primes. Seventeen is the sixth Mersenne prime exponent, yielding 131,071. Seventeen is the only prime number which is the sum of four consecutive primes: 2,3,5,7. Any other four consecutive primes summed would always produce an even number, thereby divisible by 2 and so not prime. Seventeen can be written in the form x^y + y^x and x^y - y^x, and, as such, it is a Leyland ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
16 (number)
16 (sixteen) is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and . In English speech, the numbers 16 and 60 are sometimes confused, as they sound very similar. Sixteen is the fourth power of two. For this reason, 16 was used in weighing light objects in several cultures. The British have 16 ounces in one pound; the Chinese used to have 16 ''liangs'' in one ''jin''. In old days, weighing was done with a beam balance to make equal splits. It would be easier to split a heap of grains into sixteen equal parts through successive divisions than to split into ten parts. Chinese Taoists did finger computation on the trigrams and hexagrams by counting the finger tips and joints of the fingers with the tip of the thumb. Each hand can count up to 16 in such manner. The Chinese abacus uses two upper beads to represent the 5s and 5 low ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
15 (number)
15 (fifteen) is the natural number following 14 and preceding 16. Mathematics 15 is: * A composite number, and the sixth semiprime; its proper divisors being , and . * A deficient number, a smooth number, a lucky number, a pernicious number, a bell number (i.e., the number of partitions for a set of size 4), a pentatope number, and a repdigit in binary (1111) and quaternary (33). In hexadecimal, and higher bases, it is represented as F. * A triangular number, a hexagonal number, and a centered tetrahedral number. * The number of partitions of 7. * The smallest number that can be factorized using Shor's quantum algorithm. * The magic constant of the unique order-3 normal magic square. * The number of supersingular primes. Furthermore, * 15 is one of two numbers within the ''teen'' numerical range (13-19) not to use a single-digit number in the prefix of its name (the first syllable preceding the ''teen'' suffix); instead, it uses the adjective form of five (' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
14 (number)
14 (fourteen) is a natural number following 13 and preceding 15. In relation to the word "four" ( 4), 14 is spelled "fourteen". In mathematics * 14 is a composite number. * 14 is a square pyramidal number. * 14 is a stella octangula number. * In hexadecimal, fourteen is represented as E * Fourteen is the lowest even ''n'' for which the equation φ(''x'') = ''n'' has no solution, making it the first even nontotient (see Euler's totient function). * Take a set of real numbers and apply the closure and complement operations to it in any possible sequence. At most 14 distinct sets can be generated in this way. ** This holds even if the reals are replaced by a more general topological space. See Kuratowski's closure-complement problem * 14 is a Catalan number. * Fourteen is a Companion Pell number. * According to the Shapiro inequality 14 is the least number ''n'' such that there exist ''x'', ''x'', ..., ''x'' such that :\sum_^ \frac < \frac where ''x'' = ''x'', ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
13 (number)
13 (thirteen) is the natural number following 12 and preceding 14. Strikingly folkloric aspects of the number 13 have been noted in various cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars (there are approximately 12.41 lunations per solar year, and hence 12 "true months" plus a smaller, and often portentous, thirteenth month). This can be witnessed, for example, in the "Twelve Days of Christmas" of Western European tradition. In mathematics The number 13 is the sixth prime number. It is a twin prime with 11, as well as a cousin prime with 17. It is the second Wilson prime, of three known (the others being 5 and 563), and the smallest emirp in decimal. 13 is: *The second star number: *The third centered square number: * A happy number and a lucky number. *A Fibonacci number, preceded by 5 and 8. *The smallest number whose fourth power can be written as a sum of two consecutive square numbers (1192 + 1202). *The s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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