# Vocabulary/qco

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`q: y`Prime Factors

Rank Infinity *-- operates on x and y as a whole --*
WHY IS THIS IMPORTANT?

The prime factorization of integer `y`, listed in ascending order.

The same as (`3 p: y`). See Primes (`p:`).

q: 2^31 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 q: 1+ 2^31 3 715827883 q: 2+ 2^31 2 5 5 13 41 61 1321

### Common uses

Mathematical investigations.

### Related Primitives

Primes (`p:`)

`x q: y`Prime Exponents

Rank 0 0 *-- operates on individual atoms of x and y, producing a result that may require fill --*
WHY IS THIS IMPORTANT?

Selected primes and their exponents in the factorization of `y`.

An exponent of `0` means that the corresponding prime is not a factor of `y`.

p: i.10 NB. the first 10 primes 2 3 5 7 11 13 17 19 23 29 q: 700 NB. prime factors of 700 2 2 5 5 7 _ q: 700 NB. 700 = (2^2) * (3^0) * (5^2) * (7^1) 2 0 2 1 __ q: 700 2 5 7 2 2 1

The rank of the result, and the values listed, depend on `x`.

`x`**Result**

**Rank****Meaning of Results****Number of Primes****Zero exponents included?**positive 1 The **leading**exponents of the prime factorization (keeping primes in ascending order)`x`(`x=_`means "up through the last nonzero exponent")yes negative 2 *Second*row: The**trailing**exponents of the prime factorization (keeping primes in ascending order)`|x`(`x=__`means "all nonzero exponents")no *First*row: the corresponding primes

### Common uses

1. Calculate the number of divisors of a number

*/ >: _ q: 17 2 */ >: _ q: 60 12

### More Information

1. When `x` is negative:

- the table
`t =. primes,:exponents`is calculated, including primes with`0`exponents, ending with the largest prime with a nonzero exponent; - The last
`(|x)<.({:$t)`columns of`t`are selected (i. e. the last`|x`columns, but no more columns than exist in`t`); - Columns with 0 exponents are then deleted.

Because the zero exponents are deleted at the last step, the result of ` x q: y ` is not the same as that of ` x {. __ q: y `.

_3 q: 2*3*5*17 17 1

2. ` __ q: y ` is equivalent to ` 2 p: y `.

### Related Primitives

Primes (`p:`)

### Details

1. The result has integer or extended-integer precision, depending on the size of the largest prime factor.