In this quick and easy guide, we'll work out whether 28 is divisible by 2. There are some simple rules we can follow to decide whether one number is divisible by another without ever needing to even do the division! Show
First up, let's clarify what we mean by "28 is divisible by 2". What we want to check is whether 28 can be divided by 2 without any remainder (i.e the answer is a whole number). There are a bunch of very similar methods for working out whether 28 is divisible by 2 but the easiest way to figure it out is to check whether 28 is an even number. If it is, then it is divisible by 2. In this case, 28 is an even number, which means that it is divisible by 2. You could also work this out by checking if the last digit of 28 ends with 2, 4, 6, 8, or 0. If it does, then it is divisible by two. Another way you can figure out if 28 is divisible by 2 is by actually doing the calculation and dividing 28 by 2: 28 / 2 = 14 Since the answer to our division is a whole number, we know that 28 is divisible by 2. Hopefully now you know exactly how to work out whether one number is divisible by another. Could I have just told you to divide 28 by 2 and check if it is a whole number? Yes, but aren't you glad you learned the process? Give this a go for yourself and try to calculate a couple of these without using our calculator. Grab a pencil and a piece of paper and pick a couple of numbers to try it with. Cite, Link, or Reference This PageIf you found this content useful in your research, please do us a great favor and use the tool below to make sure you properly reference us wherever you use it. We really appreciate your support!
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Illustration of the perfect number status of the number 6 In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its
aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby is an even perfect number whenever is a prime of the form for positive integer —what is now called a Mersenne prime. Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form.[1] This is known as the Euclid–Euler theorem. It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first few perfect numbers are 6, 28, 496 and 8128 (sequence A000396 in the OEIS). History[edit]In about 300 BC Euclid showed that if 2p − 1 is prime then 2p−1(2p − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.[2] In modern language, Nicomachus states without proof that every perfect number is of the form where is prime.[3][4] He seems to be unaware that n itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.) Philo of Alexandria in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by Origen,[5] and by Didymus the Blind, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14–19).[6] St Augustine defines perfect numbers in City of God (Book XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician Ismail ibn Fallūs (1194–1252) mentioned the next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) and listed a few more which are now known to be incorrect.[7] The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician.[8] In 1588, the Italian mathematician Pietro Cataldi identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.[9][10][11] Even perfect numbers[edit]Unsolved problem in mathematics: Are there infinitely many perfect numbers? Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Elements, Prop. IX.36). For example, the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number, as follows: for p = 2: 21(22 − 1) = 2 × 3 = 6for p = 3: 22(23 − 1) = 4 × 7 = 28for p = 5: 24(25 − 1) = 16 × 31 = 496for p = 7: 26(27 − 1) = 64 × 127 = 8128.Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. For 2p − 1 to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2p − 1 with a prime p are prime; for example, 211 − 1 = 2047 = 23 × 89 is not a prime number.[12] In fact, Mersenne primes are very rare—of the 2,610,944 prime numbers p up to 43,112,609,[13] 2p − 1 is prime for only 47 of them. Although Nicomachus had stated (without proof) that all perfect numbers were of the form where is prime (though he stated this somewhat differently), Ibn al-Haytham (Alhazen) circa AD 1000 conjectured only that every even perfect number is of that form.[14] It was not until the 18th century that Leonhard Euler proved that the formula 2p−1(2p − 1) will yield all the even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem. An exhaustive search by the GIMPS distributed computing project has shown that the first 48 even perfect numbers are 2p−1(2p − 1) for p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609 and 57885161 (sequence A000043 in the OEIS).[15]Three higher perfect numbers have also been discovered, namely those for which p = 74207281, 77232917, and 82589933. Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for p below 109332539. As of December 2018, 51 Mersenne primes are known,[16] and therefore 51 even perfect numbers (the largest of which is 282589932 × (282589933 − 1) with 49,724,095 digits). It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes. As well as having the form 2p−1(2p − 1), each even perfect number is the (2p − 1)th triangular number (and hence equal to the sum of the integers from 1 to 2p − 1) and the 2p−1th hexagonal number. Furthermore, each even perfect number except for 6 is the ((2p + 1)/3)th centered nonagonal number and is equal to the sum of the first 2(p−1)/2 odd cubes (odd cubes up to the cube of 2(p+1)/2-1): Even perfect numbers (except 6) are of the form with each resulting triangular number T7 = 28, T31 = 496, T127 = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T2 = 3, T10 = 55, T42 = 903, T2730 = 3727815, ...[17] This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 + 2 + 8 = 19, 1 + 9 = 10, and 1 + 0 = 1. This works with all perfect numbers 2p−1(2p − 1) with odd prime p and, in fact, with all numbers of the form 2m−1(2m − 1) for odd integer (not necessarily prime) m. Owing to their form, 2p−1(2p − 1), every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example, 610 = 22 + 21 = 1102,2810 = 24 + 23 + 22 = 111002,49610 = 28 + 27 + 26 + 25 + 24 = 1111100002, and812810 = 212 + 211 + 210 + 29 + 28 + 27 + 26 = 11111110000002.Thus every even perfect number is a pernicious number. Every even perfect number is also a practical number (cf. Related concepts). Odd perfect numbers[edit]Unsolved problem in mathematics: Are there any odd perfect numbers? It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers,[18] thus implying that no odd perfect number exists. Euler stated: "Whether ... there are any odd perfect numbers is a most difficult question".[19] More recently, Carl Pomerance has presented a heuristic argument suggesting that indeed no odd perfect number should exist.[20] All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1. Many of the properties proved about odd perfect numbers also apply to Descartes numbers, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.[21] Any odd perfect number N must satisfy the following conditions:
Furthermore, several minor results are known about the exponents e1, ..., ek.
In 1888, Sylvester stated:[46]
Minor results[edit]All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's strong law of small numbers:
[edit]The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number. By definition, a perfect number is a fixed point of the restricted divisor function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also -perfect numbers, or Granville numbers. A semiperfect number is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers. See also[edit]
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What is an example of deductive reasoning?With this type of reasoning, if the premises are true, then the conclusion must be true. Logically Sound Deductive Reasoning Examples: All dogs have ears; golden retrievers are dogs, therefore they have ears. All racing cars must go over 80MPH; the Dodge Charger is a racing car, therefore it can go over 80MPH.
What is a deductive statement?Deductive reasoning is a logical process in which a conclusion is based on the concordance of multiple premises that are generally assumed to be true. Deductive reasoning is sometimes referred to as top-down logic. Deductive reasoning relies on making logical premises and basing a conclusion around those premises.
What are examples of inductive reasoning?What is inductive reasoning?. What is an example of inductive and deductive reasoning?Inductive Reasoning: Most of our snowstorms come from the north. It's starting to snow. This snowstorm must be coming from the north. Deductive Reasoning: All of our snowstorms come from the north.
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