Divisors and multiples

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Divisors and multiples

If x * y = z (x, y, z are integers), we can define the operator / as z / y = x. x * 0 = 0, however, we can not determine 0 / 0 to one value, so that we can not define it.

Also, in the case of x * y = z (x, y, z are integers), we call z a multiple of x, and we call x a divisor of z.

To find all positive divisors of a positive integer n, you may examine whether it is a divisor of n for all integers from 1 to n.
Example: Click the "..." button.
Positive multiples of are and so on.
Positive divisors of are .

Primes

Now we think about divisors of a positive integer n. 1 and n are divisors of n. A positive integer of 2 and over which has no divisors without 1 or n is called a prime.

To examine whether a positive integer n (of 2 and over) is a prime, you may examine whether it is a divisor of n for all integers from 1 to n. If divisors are only 1 and n, n is a prime.
Example: Click the "..." button.
Primes of or less are
.

A positive integer can be shown as a product of primes.

Example: Click the "..." button.
can be shown as
.
To find the way to show a positive integer n as a product of primes, you may examine "how many time does it divide n" for all primes (you may do effectively all primes which divide n).
Usage:
(1) Click the "start" button.
(2) Click the "<-" and "->" button, change the prime to multiply, and click the "*" button, and repeat this way. (To return it, click the "/" button)

Prime:
holds.
We can find all positive divisors of a positive integer n by using the way of showing n as a product of primes.
Example: Because 50 = 2 * 5 * 5, 1, 5, 5 * 5 = 25, 2, 2 * 5 = 10, 2 * 5 * 5 = 50 are divisors of 50. Also, we can find they are only divisors of 50.
Example: Click the "..." button.
Because all positive divisors of are
,
holds.

The greatest common divisor and the least common multiple

A common divisor of two integers (three, four, five ... are possible but we think about only the case of two here) is a common divisor of the two integers. A common multiple of two integers is a common multiple of the two integers. The greatest common divisor is the greatest of all positive common divisors.
But "the maximum in the common divisor" must be positive.
The least common multiple is the least in the positive common multiples.
There are infinite positive common multiples, so it is the least of infinite things.
Example: Click the "Calc" button.

Positive multiples of are and so on.
Positive multiples of are and so on.
Positive common multiples of and are and so on.
The least common multiple of and is .

Positive divisors of are .
Positive divisors of are .
Positive common divisors of and are .
The greatest common divisor of and is .
Using the way of showing a positive integer as a product of primes, we can find the greatest common divisor and the least common multiple.
Example: Click the "Calc" button.

Positive divisors of are
, therefore
.
Positive divisors of are
, therefore
.

Positive common divisors of
are
.
These are all of positive divisors of
.
The greatest common divisor of
and
is
.
The least common multiple of
and
is
.