subclass_of sphenic number
A prime ___MASK7___ ( or a prime ) is a ___MASK86___ greater than 1 that has no positive divisors other than 1 and itself . A ___MASK86___ greater than 1 that is not a prime ___MASK7___ is called a composite ___MASK7___ . For example , 5 is prime because 1 and 5 are its only positive integer factors , whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6 . The fundamental theorem of arithmetic establishes the central role of primes in ___MASK7___ theory : any integer greater than 1 can be expressed as a product of primes that is unique up to ordering . The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization , e . g ., 3 , 1 · 3 , 1 · 1 · 3 , etc . are all valid factorizations of 3 . In ___MASK85___ , a ___MASK66___ is any number that can be expressed as the quotient or fraction " p "/" q " of two integers , a numerator " p " and a non - zero denominator " q ". Since " q " may be equal to 1 , every integer is a ___MASK66___ . The set of all rational numbers , often referred to as " the rationals ", is usually denoted by a boldface Q ( or blackboard bold formula_1 , Unicode  ); it was thus denoted in 1895 by Giuseppe Peano after " quoziente ", Italian for " quotient ". Broadly speaking , pure ___MASK85___ is ___MASK85___ that studies entirely abstract concepts . This was a recognizable ___MASK40___ of mathematical activity from the 19th century onwards , at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering , and so on . In ___MASK7___ theory , a sphenic ___MASK7___ ( from Ancient Greek :  , ' wedge ' ) is a positive integer that is the product of three distinct prime numbers . The smallest sphenic ___MASK7___ is 30 = 2 × 3 × 5 , the product of the smallest three primes . Note that this definition is more stringent than simply requiring the integer to have exactly three prime factors ; e . g . 60 = 22 × 3 × 5 has exactly 3 prime factors , but is not sphenic . All sphenic numbers have exactly eight divisors . If we express the sphenic ___MASK7___ as n = p \ cdot q \ cdot r , where p , q , and r are distinct primes , then the set of divisors of n will be : \ left \ ( 1 , \ p , \ q , \ r , \ pq , \ pr , \ qr , \ n \ right \ ) . All sphenic numbers are by definition squarefree , because the prime factors must be distinct . The Möbius function of any sphenic ___MASK7___ is  1 . The cyclotomic polynomials \ Phi_n ( x ) , taken over all sphenic numbers n , may contain arbitrarily large coefficients ( for n a product of two primes the coefficients are \ pm 1 or 0 ) . The first few sphenic numbers are : 30 , 42 , 66 , 70 , 78 , 102 , 105 , 110 , 114 , 130 , 138 , 154 , 165 , ... ( sequence A007304 in OEIS ) As of January 2016 the largest known sphenic ___MASK7___ is ( 274 , 207 , 281  1 ) × ( 257 , 885 , 161  1 ) × ( 243 , 112 , 609  1 ) , i . e . , the product of the three largest known primes . An ___MASK56___ ( from the ___MASK68___ " ___MASK56___ " meaning " whole ") is a ___MASK7___ that can be written without a fractional component . For example , 21 , 4 , 0 , and  2048 are integers , while 9 . 75 , , and are not . In ___MASK7___ theory , the prime factors of a positive integer are the prime numbers that divide that integer exactly . The prime factorization of a positive integer is a list of the integer ' s prime factors , together with their multiplicities ; the process of determining these factors is called integer factorization . The fundamental ___MASK32___ of arithmetic says that every positive integer has a single unique prime factorization . ___MASK7___ theory or , in older usage , arithmetic is a branch of pure mathematics devoted primarily to the ___MASK48___ of the integers . It is sometimes called " The Queen of Mathematics " because of its foundational place in the discipline . ___MASK7___ theorists ___MASK48___ prime numbers as well as the properties of objects made out of integers ( e . g ., rational numbers ) or defined as generalizations of the integers ( e . g ., algebraic integers ). In ___MASK38___ theory , an algebraic integer is a ___MASK28___ that is a root of some monic polynomial ( a polynomial whose leading coefficient is 1 ) with coefficients in formula_1 ( the set of integers ). The set of all algebraic integers , , is closed under addition and multiplication and therefore is a commutative subring of the complex numbers . The ___MASK47___ is the integral ___MASK29___ of regular integers formula_1 in complex numbers .