# Definition:Radical of Integer

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## Definition

The **radical of an integer** $n \in \Z$ is the product of the individual prime factors of $n$.

The radicals of the first few integers are given here:

$n$ | Decomposition | $\map {\operatorname {rad} } n$ | ||
---|---|---|---|---|

$1$ | $1$ | $1$ | ||

$2$ | $2$ | $2$ | ||

$3$ | $3$ | $3$ | ||

$4$ | $2^2$ | $2$ | ||

$5$ | $5$ | $5$ | ||

$6$ | $2 \times 3$ | $6$ | ||

$7$ | $7$ | $7$ | ||

$8$ | $2^3$ | $2$ | ||

$9$ | $3^2$ | $3$ | ||

$10$ | $2 \times 5$ | $10$ | ||

$11$ | $11$ | $11$ | ||

$12$ | $2^2 \times 3$ | $6$ | ||

$13$ | $13$ | $13$ | ||

$14$ | $2 \times 7$ | $14$ | ||

$15$ | $3 \times 5$ | $15$ | ||

$16$ | $2^4$ | $2$ |

This sequence is A007947 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The radical of $n$ can alternatively be described as the largest square-free integer which divides $n$.

## Also known as

The **radical** of an integer is also known as the **square-free kernel**.

## Also see