Compute Divisors for Positive Integers

Use the form below to get the divisors of and additional information about any positive integer up to 1 million. (decimal places are rounded)

Or just append the integer you want information about to the end of this site's URL. e.g., www.positiveintegers.org/5

   

The Integers 1 to 10000

  • Range is a range of numbers, in groups of 100. Click on the range for more information about that range.
  • Count(Primes) is the count of Prime Numbers in that range.
  • Count(Fibonacci) is the count of Fibonacci Numbers in that range.
  • Max(Count(d(N))) is the highest number of divisors that any single number within that range possesses.
  • Most Composite N is the list of the numbers in the range that have the most divisors.
  • Count(Deficient), Count(Abundant), and Count(Perfect) are the counts of Deficient, Abundant, and Perfect numbers in that range.
Range Count(Primes) Count(Fibonacci) Max(Count(d(N))) Most Composite N Count(Deficient) Count(Abundant) Count(Perfect)
1-100 25 10 12 60, 72, 84, 90, 96 76 22 2
101-200 21 1 18 180 76 24 0
201-300 16 1 20 240 77 23 0
301-400 16 1 24 360 73 27 0
401-500 17 0 24 420, 480 74 25 1
501-600 14 0 24 504, 540, 600 76 24 0
601-700 16 1 24 630, 660, 672 76 24 0
701-800 14 0 30 720 74 26 0
801-900 15 0 32 840 75 25 0
901-1000 14 1 28 960 74 26 0
1001-1100 16 0 32 1080 77 23 0
1101-1200 12 0 30 1200 76 24 0
1201-1300 15 0 36 1260 76 24 0
1301-1400 11 0 32 1320 74 26 0
1401-1500 17 0 36 1440 74 26 0
1501-1600 12 1 32 1512, 1560 77 23 0
1601-1700 15 0 40 1680 74 26 0
1701-1800 12 0 36 1800 75 25 0
1801-1900 12 0 32 1848, 1890 76 24 0
1901-2000 13 0 36 1980 74 26 0
2001-2100 14 0 36 2016, 2100 74 26 0
2101-2200 10 0 40 2160 76 24 0
2201-2300 15 0 32 2280 75 25 0
2301-2400 15 0 36 2340, 2400 77 23 0
2401-2500 10 0 30 2448 74 26 0
2501-2600 11 1 48 2520 74 26 0
2601-2700 15 0 40 2640 78 22 0
2701-2800 14 0 36 2772 74 26 0
2801-2900 12 0 42 2880 75 25 0
2901-3000 11 0 36 2940 74 26 0
3001-3100 12 0 40 3024 76 24 0
3101-3200 10 0 40 3120 76 24 0
3201-3300 11 0 40 3240 74 26 0
3301-3400 15 0 48 3360 74 26 0
3401-3500 11 0 36 3420 74 26 0
3501-3600 14 0 45 3600 77 23 0
3601-3700 13 0 40 3696 78 22 0
3701-3800 12 0 48 3780 73 27 0
3801-3900 11 0 36 3840, 3900 74 26 0
3901-4000 11 0 48 3960 75 25 0
4001-4100 15 0 42 4032 74 26 0
4101-4200 9 1 48 4200 76 24 0
4201-4300 16 0 36 4284 74 26 0
4301-4400 9 0 48 4320 77 23 0
4401-4500 11 0 36 4410, 4500 77 23 0
4501-4600 12 0 40 4536, 4560 73 27 0
4601-4700 12 0 48 4620, 4680 74 26 0
4701-4800 12 0 42 4800 74 26 0
4801-4900 8 0 36 4860, 4896 76 24 0
4901-5000 15 0 36 4950 77 23 0
5001-5100 12 0 60 5040 75 25 0
5101-5200 11 0 36 5148 74 26 0
5201-5300 10 0 48 5280 77 23 0
5301-5400 10 0 48 5400 74 26 0
5401-5500 13 0 48 5460 75 25 0
5501-5600 13 0 48 5544 74 26 0
5601-5700 12 0 40 5616, 5670 75 25 0
5701-5800 10 0 48 5760 76 24 0
5801-5900 16 0 48 5880 73 27 0
5901-6000 7 0 48 5940 74 26 0
6001-6100 12 0 48 6048 76 24 0
6101-6200 11 0 48 6120 77 23 0
6201-6300 13 0 54 6300 72 28 0
6301-6400 15 0 42 6336 75 25 0
6401-6500 8 0 50 6480 74 26 0
6501-6600 11 0 48 6552, 6600 76 24 0
6601-6700 10 0 36 6624, 6660 73 27 0
6701-6800 12 1 56 6720 76 24 0
6801-6900 12 0 48 6840 76 24 0
6901-7000 13 0 48 6930 74 26 0
7001-7100 9 0 48 7020 76 24 0
7101-7200 10 0 54 7200 74 26 0
7201-7300 11 0 40 7280 74 26 0
7301-7400 9 0 48 7392 76 24 0
7401-7500 11 0 42 7488 75 25 0
7501-7600 15 0 64 7560 73 27 0
7601-7700 12 0 40 7680 76 24 0
7701-7800 10 0 48 7800 77 23 0
7801-7900 10 0 36 7812, 7840 75 25 0
7901-8000 10 0 60 7920 74 26 0
8001-8100 11 0 48 8064 72 28 0
8101-8200 10 0 48 8160, 8190 78 21 1
8201-8300 14 0 48 8280 74 26 0
8301-8400 9 0 60 8400 77 23 0
8401-8500 8 0 40 8424 73 27 0
8501-8600 12 0 48 8568, 8580 74 26 0
8601-8700 13 0 56 8640 78 22 0
8701-8800 11 0 48 8736 75 25 0
8801-8900 13 0 54 8820 75 25 0
8901-9000 9 0 48 9000 76 24 0
9001-9100 11 0 50 9072 76 24 0
9101-9200 12 0 48 9120, 9180 73 27 0
9201-9300 11 0 64 9240 74 26 0
9301-9400 11 0 60 9360 75 25 0
9401-9500 15 0 48 9450 75 25 0
9501-9600 7 0 48 9504, 9576, 9600 75 25 0
9601-9700 13 0 48 9660 76 24 0
9701-9800 11 0 48 9720 73 27 0
9801-9900 12 0 54 9900 78 22 0
9901-10000 9 0 40 9936 77 23 0

The Integers

The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (-1, -2, -3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z - The Integers in blackboard bold, which stands for Zahlen (German for "numbers").

Positive Integers refers to all whole number greater than zero. Zero is not a positive integer. For each positive integer there is a negative integer. Integers greater than zero are said to have a positive “sign”.

The Positive Integers are a subset of the Natural Numbers (N - The Natural Numbers), depending on whether or not 0 is considered a Natural Number. The term Positive Integers is preferred over Natural Numbers and Counting Numbers because it is more clearly defined; there is inconsistency over whether zero is a member of those sets. Zero is not an element of the Positive Integers.

The Positive Integers are symbolized by Z+ - The Positive Integers.

Prime numbers are a subset of the positive integers and are of special interest in Number Theory. Note that the number 1 is not a prime number; i.e., for the set of prime numbers P - The Prime Numbers, all P - The Prime Numbers > 1. A prime number is a positive integer that has no positive integer divisors except for 1 and itself. Positive Integers that are not Prime Numbers or 1 are Composite Numbers. The number 1 is neither a Prime Number nor a Composite Number.

Algebraic properties of Integers

Like the natural numbers, Z - The Integers is closed under the operations of addition and multiplication; that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers and zero, Z - The Integers (unlike the natural numbers) is also closed under subtraction. Z - The Integers is not closed under the operation of division, since the quotient of two integers ( e.g. , 1 divided by 2), need not be an integer.

The following table lists some of the basic properties of addition and multiplication for any integers a , b and c .

addition multiplication
closure : a  +  b    is an integer a  ×  b    is an integer
associativity : a  + ( b  +  c )  =  ( a  +  b ) +  c a  × ( b  ×  c )  =  ( a  ×  b ) ×  c
commutativity : a  +  b   =   b  +  a a  ×  b   =   b  ×  a
existence of an identity element : a  + 0  =   a a  × 1  =   a
existence of inverse elements : a  + (- a )  =  0
distributivity : a  × ( b  +  c )  =  ( a  ×  b ) + ( a  ×  c )

Ordering

Z - The Integers is a totally ordered set without an upper or lower bound. The ordering of Z - The Integers is given by

... < -2 < -1 < 0 < 1 < 2 < ...

An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:

  1. if a < b and c < d , then a + c < b + d
  2. if a < b and 0 < c , then ac < bc

Divisors

A divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. If x is a divisor of n, it can be written that x|n. This is read as x divides n. It is also said that n is divisible by x, and that n is a multiple of x.

1 and -1 are divisors of every integer, and every integer is a divisor of 0. Numbers divisible by 2 are called even and those that are not are called odd.

The name comes from the arithmetic operation of division : if a/b=c, then a is the dividend, b the divisor, and c the quotient.

Some elementary properties of Divisors are:

  • If a|b and a|c, then a|(b+c).
  • If a|b and b|c, then a|c.
  • If a|b and b|a, then a=b or a=-b.

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