540radical

540radical refers to the radical of the integer 540 in number theory, denoted rad(540). The radical of a positive integer n is defined as the product of its distinct prime factors. If n = p1^a1 p2^a2 ... pk^ak, then rad(n) = p1 p2 ... pk. For 540, the prime factorization is 540 = 2^2 × 3^3 × 5, so the distinct primes are 2, 3 and 5, and rad(540) = 2 × 3 × 5 = 30.

Properties of the radical function include that rad(n) is always squarefree (no repeated prime factors) and rad(n) divides n. The function is multiplicative for coprime arguments, meaning rad(ab) = rad(a) rad(b) when gcd(a, b) = 1. In general the exponents in n do not affect rad(n), only which primes appear in n.

Context and applications: the radical function rad(n) appears in various areas of number theory, particularly in multiplicative function studies and conjectures such as the ABC conjecture, where rad(abc) involves the product of distinct primes dividing abc. It provides a way to measure the prime support of an integer independent of powers.

Examples: rad(240) = 2 × 3 × 5 = 30, since 240 = 2^4 × 3 × 5. rad(1) is defined as 1. The concept is a standard tool for understanding the prime structure of integers in theoretical discussions and problem solving. See also radical (number theory), rad(n), and related multiplicative functions.