Composite digits

The only single digit composite numbers are $4, 6, 8,$ and $9.$ Then the number we are looking for obviously ends in $9$ ; it is divisible by $2$ otherwise. So checking the first few possibilities, $49 = 7^2 \text{ (not prime)} \\ 69 = 3\times 23 \text{ (not prime)} \\ \boxed{89\text{ is prime}}$ and is therefore the smallest prime number with composite digits.

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public static void main(String args[]){
    for(int i = 2;; i++)
        if(prime(i) && chk(Integer.toString(i)))
            System.out.println(i);
}

static boolean prime(int n) {
    if(n == 2) return true;
    if (n % 2 == 0) return false;
    for(int i = 3; i * i <= n; i += 2)
        if(n % i == 0) return false;
    return true;
}

static boolean chk(String n){
    boolean m = true;
    for(int i = 0; i < n.length(); i++)
        if(prime(Character.getNumericValue(n.charAt(i)))){
            m = false;
            break;
        }
    return m;
}

Execution time ~ 0.00028 seconds