Is 41 A Composite Number

Positive integer having at least 1 divisor other than 1 or itself

Demonstration, with Cuisenaire rods, of the divisors of the composite number 10

Comparison of prime and blended numbers

A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself.^{[one] [2]} Every positive integer is composite, prime number, or the unit one, so the composite numbers are exactly the numbers that are non prime and non a unit.^{[3] [4]}

For example, the integer 14 is a composite number because information technology is the production of the ii smaller integers 2 × seven. Likewise, the integers two and 3 are non composite numbers because each of them can just be divided by one and itself.

The blended numbers up to 150 are:

4, 6, eight, 9, 10, 12, 14, fifteen, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, xxx, 32, 33, 34, 35, 36, 38, 39, twoscore, 42, 44, 45, 46, 48, 49, fifty, 51, 52, 54, 55, 56, 57, 58, lx, 62, 63, 64, 65, 66, 68, 69, seventy, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, xc, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150. (sequence A002808 in the OEIS)

Every blended number can be written as the product of ii or more than (not necessarily singled-out) primes.^[5] For instance, the composite number 299 can be written as xiii × 23, and the composite number 360 can be written as twoⁱⁱⁱ × 3^two × five; furthermore, this representation is unique upward to the order of the factors. This fact is called the central theorem of arithmetic.^{[half-dozen] [7] [viii] [nine]}

In that location are several known primality tests that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input.

Types [edit]

One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-virtually prime (the factors need not be distinct, hence squares of primes are included). A composite number with iii singled-out prime number factors is a sphenic number. In some applications, it is necessary to differentiate betwixt composite numbers with an odd number of distinct prime number factors and those with an even number of singled-out prime factors. For the latter

${\displaystyle \mu (northward)=(-1)^{2x}=1}$

(where μ is the Möbius function and x is half the total of prime factors), while for the former

${\displaystyle \mu (n)=(-i)^{2x+1}=-ane.}$

However, for prime numbers, the function also returns −ane and ${\displaystyle \mu (1)=1}$ . For a number due north with 1 or more than repeated prime factors,

${\displaystyle \mu (n)=0}$ .^[10]

If all the prime factors of a number are repeated information technology is called a powerful number (All perfect powers are powerful numbers). If none of its prime factors are repeated, information technology is called squarefree. (All prime numbers and i are squarefree.)

For case, 72 = 2ⁱⁱⁱ × three², all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × three × 7, none of the prime factors are repeated, so 42 is squarefree.

Another way to allocate blended numbers is by counting the number of divisors. All composite numbers have at least 3 divisors. In the example of squares of primes, those divisors are ${\displaystyle \{1,p,p^{2}\}}$ . A number north that has more than divisors than any ten < n is a highly composite number (though the starting time two such numbers are 1 and ii).

Composite numbers accept also been chosen "rectangular numbers", but that proper noun can also refer to the pronic numbers, numbers that are the product of two sequent integers.

Yet another way to classify composite numbers is to determine whether all prime number factors are either all beneath or all in a higher place some fixed (prime) number. Such numbers are called smooth numbers and rough numbers, respectively.

Run into also [edit]

• Canonical representation of a positive integer
• Integer factorization
• Sieve of Eratosthenes
• Tabular array of prime factors

Notes [edit]

1. ^ Pettofrezzo & Byrkit (1970, pp. 23–24)
2. ^ Long (1972, p. 16)
3. ^ Fraleigh (1976, pp. 198, 266)
4. ^ Herstein (1964, p. 106)
5. ^ Long (1972, p. 16)
6. ^ Fraleigh (1976, p. 270)
7. ^ Long (1972, p. 44)
8. ^ McCoy (1968, p. 85)
9. ^ Pettofrezzo & Byrkit (1970, p. 53)
10. ^ Long (1972, p. 159)

References [edit]

• Fraleigh, John B. (1976), A First Class In Abstract Algebra (second ed.), Reading: Addison-Wesley, ISBN0-201-01984-1
• Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN978-1114541016
• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Visitor, LCCN 77-171950
• McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766

External links [edit]

• Lists of composites with prime number factorization (first 100, one,000, x,000, 100,000, and 1,000,000)
• Divisor Plot (patterns establish in large composite numbers)

Is 41 A Composite Number,

Source: https://en.wikipedia.org/wiki/Composite_number

Posted by: heinenalwaroullace.blogspot.com

Share this post