Divisors of a Number

The integer $ b $ (non-zero $ b \in \mathbb{N}_{>0} $) is a divisor of the integer $ a $ ($ \in \mathbb{N} $) if there is a integer $ c $ ($ \in \mathbb{N} $) such that $ c = a/b $ (NB: $ c $ is an integer, without decimal part).

In this case, $ c $ is represented as a division of $ a $ by $ b $ so $ b $ is indeed a divisor of $ a $ ($ a $ is divisible by $ b $).

By equivalence, $ a $ can be represented as a multiplication of $ b $ and $ c $: $ a = b \times c $, so $ a $ is a multiple of $ b $ and $ c $, and therefore $ b $ and $ c $ are divisors of $ a $.

An easy method consists in testing all numbers $ n $ between $ 1 $ and $ \sqrt{N} $ (square root of $ N $ ) to see if the remainder is equal to $ 0 $.

Another method calculates the prime factors decomposition of $ N $ and by combination of them, get all divisors.

Another?: use the form on top of this page to get the list of divisors of any other number.

The divisibility criteria are a roundabout way to know if a number is divisible by another without directly doing the calculation. Here is a (non-exhaustive) list of the main divisibility criteria (in base 10):

— Criterion of divisibility by $ 1 $: any integer number is divisible by $ 1 $

— Criterion of divisibility by $ 2 $: any number multiple of $ 2 $ has an even digit for the units digit, so the last digit is either $ 0 $ or $ 2 $ or $ 4 $ or $ 6 $ or $ 8 $.

— Criterion of divisibility by $ 3 $: any number multiple of $ 3 $ has for sum of digits a number which is also multiple of $ 3 $, and therefore the digital root of the number is $ 0 $ or $ 3 $ or $ 6 $ or $ 9 $

— Criterion of divisibility by $ 4 $: any number multiple of $ 4 $ has as the sum of the units digit and the double of the tens digit a number also divisible by 4. (Variant) the last 2 digits (tens and ones) of any number multiple of $ 4 $ are divisible by $ 4 $ (so by $ 2 $ then again by $ 2 $)

— Criterion of divisibility by $ 5 $: any number multiple of $ 5 $ has for digit of the units $ 0 $ or $ 5 $

— Criterion of divisibility by $ 6 $: any number multiple of $ 6 $ validates the criteria of divisibility by $ 2 $ and by $ 3 $

— Criterion of divisibility by $ 7 $: any number multiple of $ 7 $ has a sum of its total number of tens (all digits except the last) and of five times its units digit also divisible by 7 (criterion to be repeated in loop)

— Criterion of divisibility by $ 8 $: any number which is multiple of $ 8 $ has for the sum of the units digit, the double of the tens digit and the quadruple of the hundreds digit a number also divisible by 8.

— Criterion of divisibility by $ 9 $: any number which is multiple of $ 9 $ has as its sum a number which is also a multiple of $ 9 $, and therefore the digital root of the number is $ 9 $.

— Criterion of divisibility by $ 10 $: any number multiple of $ 10 $ has as last digit $ 0 $.

Note N the number,

Initialize the list of divisors

For i equal to 2 up to the root of N,

Attempt to divide N by i

If the remainder of the division is 0, then append i to the list of divisors

End for

Return list of divisors

Numbers that have only 2 divisors are prime numbers. They have $ 1 $ and themselves as divisors.

Numbers having exactly 3 divisors are perfect squares of prime numbers: 4, 9, 25, 49, etc.

Numbers having exactly 5 divisor are numbers of the form $ a^4 $ with $ a $ a prime number.

The number $ 0 $ has an infinity of divisors, because all the numbers divide $ 0 $ and the result is worth $ 0 $ (except for $ 0 $ itself because the division by $ 0 $ does not make sense, it is however possible to say that $ 0 $ is a multiple of $ 0 $).

$$ \frac{0}{n} = 0, (n \neq 0) $$

A negative integer has the same divisors as its positive opposite. The divisors of $ \pm N $ are the divisors of $ N $.

Technically if $ d $ is a divisor of $ N $ then $ -d $ is also a divisor of $ N $, to avoid trivial repetition, negative divisors are ignored.

The number 1 divides all numbers.

By equivalence, all integer numbers are multiples of 1.

Definition: A perfect number is a natural number N whose sum of divisors (excluding N) is equal to N.

Definition: An abundant number is a natural number $ N $ whose sum of divisors (excluding $ N $) is superior to $ N $.

Definition: a superabundant number is a number that has more divisors than any number smaller than it.

The first abundant numbers are: 1 (1 divisor), 2 (2 divisors), 4 (3 divisors), 6 (4 divisors), 12 (6 divisors), 24 (8 divisors), 36 (9 divisors), 48 (10 divisors), 60 (12 divisors), 120 (16 divisors), 180 (18 divisors), 240 (20 divisors), 360 (24 divisors), 720 (30 divisors), 840 (32 divisors), 1260 (36 divisors), 1680 (40 divisors), 2520 (48 divisors), 5040 (60 divisors), 10080 (72 divisors), 15120 (80 divisors), 25200 (90 divisors), 27720 (96 divisors), 55440 (120 divisors), 110880 (144 divisors), 166320 (160 divisors), 277200 (180 divisors), 332640 (192 divisors), 554400 (216 divisors), 665280 (224 divisors), 720720 (240 divisors), 1441440 (288 divisors), 2162160 (320 divisors), 3603600 (360 divisors), 4324320 (384 divisors), 7207200 (432 divisors), 8648640 (448 divisors), 10810800 (480 divisors), 21621600 (576 divisors)

Definition: A deficient number is a natural number N whose sum of divisors (excluding N) is inferior to N.

Two numbers are amicable if the sum of their divisors is the same and the sum of the two numbers is equal to the sum of their divisors.

The least common multiple (LCM) is the smallest number that has for divisors a list of given numbers.