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integer as a product rewriter

world's simplest integer tool

This browser-based utility rewrites an integer as a product of several integer factors and prints the multiplicative decomposition in the output. You can find all possible products or just a specific number of products. You can change the order of multipliers and exclude the trivial factor "1". You can also change the multiplication symbol and use a custom output format for the factors. Created by math nerds from team Browserling.

᠎᠎᠎          Tool Options

Products and Factors

How many integer products
to find?
Allow using the trivial
factor "1" in integer
products?

Multiplication Characters

Enter the symbol you want to
use as the multiplication symbol.
Insert this symbol between
products. (Newline by default).
Open product
character.
Close product
character.

Factor Sorting

Print the largest factor on
the right.
Print the largest factor on
the left.
Print integer factors in
mixed order.

What is a integer as a product rewriter?

With this online application, you can disassemble an integer into a bunch of smaller integers that when multiplied together produce the original integer. This is done by finding all possible divisors of the given integer and then combining them in all possible ways so that their product equals the dividend. For example, if you enter the integer 20 as input data, the program will calculate its divisors, which are 1, 2, 4, 5, 10, 20. Then, it will combine and multiply these factors and find which ones create 20. These combinations are 1 × 20, 1 × 2 × 10, 1 × 4 × 5, 1 × 2 × 2 × 5. The multiplier "1" (also called the trivial multiplier) has no effect on the product as anything multiplied by one equals the same value. You can turn off the trivial multiplier in the options and avoid using "1". In this case, you'll get only three products: 2 × 10, 4 × 5, 2 × 2 × 5. If the input integer is an integer prime, for example, 7, then it has only two divisors, 1 and 7, and can be expressed in only one way as 1 × 7. According to the commutative property of multiplication, the product 1 × 7 is the same as 7 × 1, so we only print one result. If the trivial divisor option is turned off, then there are no results. You can change the order of the multipliers and sort them in increasing order, decreasing order, or random order. The number of multiplicative partitions depends on the number of divisors and how well they combine together. For example, for the integer 8192, there are 101 different products but for the integer 8193, there are only 2. To avoid unpredictably long calculations for large integers, we limit the number of products to 100, but you can increase or decrease this amount in the options. You can also customize the output format of products. For example, you can replace the default multiplying character "×" with "*", or "·", or any other. If you're doing programming, then you can also put the products in brackets, for example, {1 × 7}, (1 × 7), or [1 × 7], and change the separator between individual products. By default, it's the newline "\n", so all products form a vertical column and each product goes on a separate line, but you can use a comma or a semicolon. Integerabulous!


Integer as a product rewriter examples

Click to try

Print All Integer Products

In this example, we decompose the integer 100 into several products that use smaller integers. The value 100 is an even integer that has nine divisors including 1 and itself. These divisors are 1, 2, 4, 5, 10, 20, 25, 50, 100. From all these terms, the algorithm creates nine combinations that multiply up to the original value 100. We display each product on a new row, arranging the factors in ascending order, and using the Unicode "×" symbol as the multiplication character.

100
1 × 100 1 × 2 × 50 1 × 4 × 25 1 × 5 × 20 1 × 10 × 10 1 × 2 × 2 × 25 1 × 2 × 5 × 10 1 × 4 × 5 × 5 1 × 2 × 2 × 5 × 5
Required options
These options will be used automatically if you select this example.
How many integer products
to find?
Allow using the trivial
factor "1" in integer
products?
Enter the symbol you want to
use as the multiplication symbol.
Insert this symbol between
products. (Newline by default).
Open product
character.
Close product
character.
Print the largest factor on
the right.

Exclude Trivial Factor

The input integer 20160 in this example, although not very large, has mighty powers. It has 84 divisors that can be combined into products to produce itself in 1260 ways (1261 ways including the trivial product 20160 × 1). Not to clutter the output with so many products, we limited the output length to just 50 products. We also disabled the option that uses the trivial divisor "1" and chose the display order to be descending (largest multipliers go first). To customize the output, we replaced the regular multiplication sign with the mid-line dot operator, put products in quoted square brackets, and separated them with the comma character.

20160
"[10080⋅2]", "[6720⋅3]", "[5040⋅4]", "[4032⋅5]", "[3360⋅6]", "[2880⋅7]", "[2520⋅8]", "[2240⋅9]", "[2016⋅10]", "[1680⋅12]", "[1440⋅14]", "[1344⋅15]", "[1260⋅16]", "[1120⋅18]", "[1008⋅20]", "[960⋅21]", "[840⋅24]", "[720⋅28]", "[672⋅30]", "[630⋅32]", "[576⋅35]", "[560⋅36]", "[504⋅40]", "[480⋅42]", "[448⋅45]", "[420⋅48]", "[360⋅56]", "[336⋅60]", "[320⋅63]", "[315⋅64]", "[288⋅70]", "[280⋅72]", "[252⋅80]", "[240⋅84]", "[224⋅90]", "[210⋅96]", "[192⋅105]", "[180⋅112]", "[168⋅120]", "[160⋅126]", "[144⋅140]", "[5040⋅2⋅2]", "[3360⋅3⋅2]", "[2520⋅4⋅2]", "[2016⋅5⋅2]", "[1680⋅6⋅2]", "[1440⋅7⋅2]", "[1260⋅8⋅2]", "[1120⋅9⋅2]", "[1008⋅10⋅2]"
Required options
These options will be used automatically if you select this example.
How many integer products
to find?
Allow using the trivial
factor "1" in integer
products?
Enter the symbol you want to
use as the multiplication symbol.
Insert this symbol between
products. (Newline by default).
Open product
character.
Close product
character.
Print the largest factor on
the left.

Random Factor Order

This example rewrites integer 608 as a product in twelve ways (there are 19 ways in total). It mixes the order of multipliers in products and includes the factor "1" in expressions. It uses the asterisk character as the multiplication symbol and the parentheses "(" and ")" as open and close product symbols. One integer product is printed per line and the factors are shuffled in a different order each time you click on the example.

608
(608 ∗ 1) (19 ∗ 32 ∗ 1) (16 ∗ 1 ∗ 38) (2 ∗ 1 ∗ 304) (1 ∗ 152 ∗ 4) (76 ∗ 8 ∗ 1) (16 ∗ 19 ∗ 1 ∗ 2) (1 ∗ 2 ∗ 76 ∗ 4) (1 ∗ 2 ∗ 38 ∗ 8) (2 ∗ 1 ∗ 2 ∗ 152) (19 ∗ 8 ∗ 4 ∗ 1) (38 ∗ 1 ∗ 4 ∗ 4) (2 ∗ 4 ∗ 4 ∗ 1 ∗ 19)
Required options
These options will be used automatically if you select this example.
How many integer products
to find?
Allow using the trivial
factor "1" in integer
products?
Enter the symbol you want to
use as the multiplication symbol.
Insert this symbol between
products. (Newline by default).
Open product
character.
Close product
character.
Print integer factors in
mixed order.

Pro tips Master online integer tools

You can pass input to this tool via ?input query argument and it will automatically compute output. Here's how to type it in your browser's address bar. Click to try!

https://onlinetools.com/integer/rewrite-integer-as-product?input=100&number-of-products=9&include-one=true&multiply-symbol=%20%25C3%2597%20&product-separator=%255Cn&open-symbol=&close-symbol=&left-to-right=true

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