Recalling Sir $Euler$

$n$ = $a^{p}$ $\times$ $b^{q}$ .....

$\phi(n)$ = $n\times(1-\dfrac{1}{a}$ )( $1-\dfrac{1}{b}$ )........

$\phi(n)$ is [ $n-1$ ] for $n$ being a prime number so largest prime number $\le$ $500$ is $\color{#3D99F6}{499}$ so

$\phi(n)_{max}$ = $\phi(499)$ = $\boxed{498}$

As we know $\phi(n)$ =n - 1 if n is prime. The the maximum value of Euler's Totient Function $\phi(n)$ for $n\le500$ , $\phi(n)$ where n is largest prime <500. n=499 then $\phi(499)$ =498.

We have $n\le500$ .

$\phi(n)$ is number of numbers less than and co-prime to $n$ . $\phi(n)$ will definitely be less than $500$ .

Let $n={a_1}^{p_1} \times {a_2}^{p_2} \times ........... \times {a_n}^{p_n}$

$\phi(n)=n\prod \left(1-\dfrac{1}{p_i}\right).$

It is evident that the more the number of factors of $n$ , the lesser is the value of $\phi(n)$ .

Thus, $n$ should have as less factors as possible, that is, $n$ should be a prime.

Largest prime less than $500$ is $499$ . $\phi(499)=\boxed{498}$ .

Euler's Totient Function of n is the number of integers x that are coprime to n. Therefore Euler's Totient Function of n is always less than n. Since 499 is the highest prime number less than 500 and there will be no Euler's Totient Function of integers less than 499 that are greater than 499, the Euler's Totient Function of 499 is the maximum value which is equal to 499 - 1 = 498

$\phi{(n)}$ stands for the number of integers $\leq n$ that are co-prime to $n.$ Now,more the number of factors of $n$ ,the lesser the number of integers that are co-prime to it and also less than it.Thus,we are looking for the number which is $\leq 500$ and has very less factors.Now,we know that a prime number has the least number of factors.Thus, $n=$ a prime number.The number of integers that are $\leq$ and co-prime to a prime $p$ are $1,2,3,4...... (p-1)$ .That is a total of $(p-1)$ numbers.Thus, $\phi(p)=p-1$ for a prime number $p.$ The largest prime $\leq 500=499.$ Thus,answer $=\phi{(499)}=498.$

Recall that the totient function of a prime p is p-1. Since 499 is prime, we have the largest value being 499-1=498

Euler Totient function for prime p is p-1 so finding the answer boils down to finding the largest prime less than 500 which in this case is 499 hence the answer is 498.