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整数環での四則演算と言うことで、素直に実装してみました。 題意にはないですが除算(乗算の逆演算)も加えてみました。答えがない場合には 0を返します。

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public class Sample115 {
    public enum Operator {
        Plus {
            @Override public String toString() { return "+"; }
            @Override public int calc(int lhs, int rhs) { return lhs + rhs; }
        },
        Minus {
            @Override public String toString() { return "-"; }
            @Override public int calc(int lhs, int rhs) { return lhs - rhs; }
        },
        Times {
            @Override public String toString() { return "*"; }
            @Override public int calc(int lhs, int rhs) { return lhs * rhs; }
        },
        Divide {
            @Override public String toString() { return "/"; }
            @Override public int calc(int lhs, int rhs) { return lhs / rhs; }
            @Override public int operate(int lhs, int rhs, int mod) {
                System.out.format("%d %s %d = ", lhs, toString(), rhs);
                int l = modulo(lhs, mod);
                int r = modulo(rhs, mod);
                if (lhs != l || rhs != r) {
                    System.out.format("%d %s %d = ", l, toString(), r);
                }
                int ret = 0;
                for (int index = 0; index < mod; index++) {
                    if (l == modulo(r * index, mod)) {
                        ret = index;
                        break;
                    }
                }
                System.out.println(ret);
                return ret;
            }
        };

        public abstract int calc(int lhs, int rhs);
        public int operate(int lhs, int rhs, int mod) {
            System.out.format("%d %s %d = ", lhs, toString(), rhs);
            int l = modulo(lhs, mod);
            int r = modulo(rhs, mod);
            if (lhs != l || rhs != r) {
                System.out.format("%d %s %d = ", l, toString(), r);
            }
            int ret = modulo(calc(l, r), mod);
            System.out.println(ret);
            return ret;
        }
        protected int modulo(int val, int mod) {
            int ret = val;
            while (ret < 0) { ret += mod; }
            while (ret >= mod) { ret -= mod; }
            return ret;
        }
    }

    public static int calcModulo(Operator op, int lhs, int rhs, int mod) {
        return op.operate(lhs, rhs, mod);
    }


    public static void main(String[] args) {
        calcModulo(Operator.Plus, 1, 2, 10);
        calcModulo(Operator.Plus, 1, 2, 3);
        calcModulo(Operator.Plus, 7, 3, 10);
        calcModulo(Operator.Plus, 11, 12, 10);
        calcModulo(Operator.Minus, 3, 2, 10);
        calcModulo(Operator.Minus, 2, 3, 10);
        calcModulo(Operator.Times, 2, 3, 10);
        calcModulo(Operator.Times, 11, 12, 10);
        calcModulo(Operator.Times, 18, 39, 10);
        calcModulo(Operator.Divide, 1, 2, 11);
        calcModulo(Operator.Divide, 2, 18, 11);
    }
}

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