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`lcm`

Least common multiple of polynomials

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Syntax

```lcm(`p`, `q, …`)
lcm(`f`, `g, …`)
```

Description

`lcm(p, q, ...)` calculates the least common multiple of any number of polynomials. The coefficient ring of the polynomials may either be the integers or the rational numbers, `Expr`, a residue class ring `IntMod(n)` with a prime number `n`, or a domain.

All polynomials must have the same indeterminates and the same coefficient ring.

Polynomial expressions are converted to polynomials. See `poly` for details. `FAIL` is returned if an argument cannot be converted to a polynomial.

The return value is of the same type as the input polynomials, i.e., either a polynomial of type `DOM_POLY` or a polynomial expression.

`lcm` returns 1 if all arguments are 1 or - 1, or if no argument is given. If at least one of the arguments is 0, then `lcm` returns 0.

Use `ilcm` if all arguments are known to be integers, since it is much faster than `lcm`.

Examples

Example 1

The least common multiple of two polynomial expressions can be computed as follows:

`lcm(x^3 - y^3, x^2 - y^2);`

One may also choose polynomials as arguments:

```p := poly(x^2 - y^2, [x, y], IntMod(17)): q := poly(x^2 - 2*x*y + y^2, [x, y], IntMod(17)): lcm(p, q)```

`delete f, g, p, q:`

Parameters

 `pq, …` `fg, …` polynomial expressions

Return Values

Polynomial, a polynomial expression, or the value `FAIL`.

` f`, ` g`, `p`, ` q`