TestClass.java

import java.util.ArrayList;
import java.util.List;

class TestClass {
  private static final int[] PRIMES = { 2, 3, 5, 7, 11, 13, 17, 19, 23 };

  public static void main(String args[]) throws Exception {
    goodVsBad(23, 46, 22);
  }

  public static void goodVsBad(int hh, int mm, int ss) {
    int total = (60 - ss) + ((59 - mm) * 60) + ((23 - hh) * 3600);
    int bads = 0;

    List<Integer> primes = null;
    for (int h = hh; h < 24; h++) {
      primes = new ArrayList<>();
      for (int i = 0; i < PRIMES.length; i++) {
        if (h % PRIMES[i] == 0) {
          primes.add(PRIMES[i]);
        }
      }

      if (h == hh)
        bads += combinations(mm, ss, primes);
      else
        bads += combinations(0, 0, primes);

    }
    
    int goods = total - bads;
    int hcf = hcf(goods, bads);

    System.out.println((bads / hcf) + ":" + (goods / hcf));
  }

  public static int combinations(int mm, int ss, List<Integer> primes) {
    return count(mm, primes) * count(ss, primes);
  }

  public static int count(int start, List<Integer> primes) {
    int count = 0;
    for (int i = start; i < 60; i++) {
      for (int p : primes) {
        if (i % p == 0) {
          count++;
          break;
        }
      }
    }
    return count;
  }

  public static int hcf(int n1, int n2) {
    int hcf = 1;
    int min = Math.min(n1, n2);

    for (int i = 1; i <= min; i++) {
      if (n1 % i == 0 && n2 % i == 0) {
        hcf = i;
      }
    }
    
    return hcf;

  }

}