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Quick suggestions...

• xIsDivisibleByY(): might be better to leave it as isDivisble(), the point being it's relatively straightforward to infer the ordering. x and y isn't much better, since they can become misleading if you decide to rename your method arguments a, b or numerator, denominator etc.

• The body of xIsDivisibleByY() can also be return y != 0 && x % y == 0.

• You can simply put xIsDivisibleByY(n + n + 1, 7) for isWow().

• Are you really expected to show your implementation for prime-checking? :D Since your upper bound is only 113, I'm kind of tempted to just use a pre-computed list of primes and call it a day...

• In any case, a slight optimization for your prime-checking can be to first test if it's divisible by 2 or not. If not, you just need to iterate through the odd numbers up to (削除) sqrtOfN (削除ここまで) threshold: a decidedly better name since sqrtOfN is never really the actual square root, is it? ;)

• As mentioned in @janos's answer, use a StringBuilder instead of rolling your own addToList() method, which is also named wrongly anyways since it isn't adding to any list.

• boolean printDirectly: It's commendable that you spot the slightly odd requirement of having to print to System.out and returning an ArrayList. However, you should go a step to further to really illustrate how your internal method is observing this 'separation-of-concerns'-like behavior as such:

   // would be great to return as a List, 
   // unfortunately caller expects an ArrayList to return itself too
   private static ArrayList<String> getResult(startInclusive, endInclusive) {
   ArrayList<String> result = new ArrayList<>();
   // ...
   return result;
   }
   public static ArrayList<String> iterate() {
   ArrayList<String> result = getResult(0, 113);
   for (String current : result) {
   System.out.println(current);
   }
   return result;
   }
  

Quick suggestions...

• xIsDivisibleByY(): might be better to leave it as isDivisble(), the point being it's relatively straightforward to infer the ordering. x and y isn't much better, since they can become misleading if you decide to rename your method arguments a, b or numerator, denominator etc.

• The body of xIsDivisibleByY() can also be return y != 0 && x % y == 0.

• You can simply put xIsDivisibleByY(n + n + 1, 7) for isWow().

• Are you really expected to show your implementation for prime-checking? :D Since your upper bound is only 113, I'm kind of tempted to just use a pre-computed list of primes and call it a day...

• In any case, a slight optimization for your prime-checking can be to first test if it's divisible by 2 or not. If not, you just need to iterate through the odd numbers up to (削除) sqrtOfN (削除ここまで) threshold: a decidedly better name since sqrtOfN is never really the actual square root, is it? ;)

• As mentioned in @janos's answer, use a StringBuilder instead of rolling your own addToList() method, which is also named wrongly anyways since it isn't adding to any list.

• boolean printDirectly: It's commendable that you spot the slightly odd requirement of having to print to System.out and returning an ArrayList. However, you should go a step to further to really illustrate how your internal method is observing this 'separation-of-concerns'-like behavior as such:

   // would be great to return as a List, 
   // unfortunately caller expects an ArrayList to return itself too
   private static ArrayList<String> getResult(startInclusive, endInclusive) {
   ArrayList<String> result = new ArrayList<>();
   // ...
   return result;
   }
   public static ArrayList<String> iterate() {
   ArrayList<String> result = getResult(0, 113);
   for (String current : result) {
   System.out.println(current);
   }
   return result;
   }