C Programming Read Natural Numbers Between 50 and 150 Display Square Root in Function

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

Sit-in, with Cuisenaire rods, of the divisors of the composite number 10

Comparison of prime and composite numbers

A blended number is a positive integer that tin can be formed past multiplying two smaller positive integers. Equivalently, information technology is a positive integer that has at to the lowest degree ane divisor other than ane and itself.^{[1] [2]} Every positive integer is composite, prime number, or the unit one, so the composite numbers are exactly the numbers that are non prime number and not a unit.^{[3] [4]}

For example, the integer 14 is a composite number because it is the product of the 2 smaller integers 2 × 7. As well, the integers 2 and three are not composite numbers because each of them can just exist divided by one and itself.

The composite numbers upward to 150 are

4, six, 8, nine, 10, 12, 14, 15, 16, xviii, xx, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, ninety, 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 composite number tin be written as the product of ii or more (not necessarily distinct) primes.^[5] For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 2^three × 3² × 5; furthermore, this representation is unique up to the club of the factors. This fact is chosen the central theorem of arithmetic.^{[half-dozen] [7] [eight] [9]}

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

Types [edit]

One fashion to classify composite numbers is past counting the number of prime factors. A composite number with 2 prime number factors is a semiprime or two-almost prime (the factors demand not be distinct, hence squares of primes are included). A blended number with 3 distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between blended numbers with an odd number of distinct prime factors and those with an fifty-fifty number of distinct prime factors. For the latter

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

(where μ is the Möbius office and ten is half the full of prime factors), while for the former

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

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

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

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

For case, 72 = 2³ × three², all the prime factors are repeated, so 72 is a powerful number. 42 = two × 3 × vii, none of the prime factors are repeated, so 42 is squarefree.

Some other style to classify blended numbers is past counting the number of divisors. All composite numbers take at to the lowest degree three divisors. In the case of squares of primes, those divisors are ${\displaystyle \{1,p,p^{2}\}}$ . A number n that has more divisors than any ten < n is a highly composite number (though the start 2 such numbers are ane and 2).

Composite numbers have also been chosen "rectangular numbers", but that name can also refer to the pronic numbers, numbers that are the production of 2 consecutive integers.

Yet another way to classify composite numbers is to make up one's mind whether all prime number factors are either all below or all above some stock-still (prime number) number. Such numbers are called smooth numbers and rough numbers, respectively.

See also [edit]

• Canonical representation of a positive integer
• Integer factorization
• Sieve of Eratosthenes
• Table of prime number factors

Notes [edit]

1. ^ Pettofrezzo & Byrkit (1970, pp. 23–24)
2. ^ Long (1972, p. xvi)
3. ^ Fraleigh (1976, pp. 198, 266)
4. ^ Herstein (1964, p. 106)
5. ^ Long (1972, p. xvi)
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 Starting time Course 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 Company, 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 factorization (outset 100, ane,000, 10,000, 100,000, and 1,000,000)
• Divisor Plot (patterns found in large composite numbers)

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

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