Make conjectures by examining if there are any patterns or relations between the parity of a number and its digital root, and the digital root of a number and the remainder obtained when the number is divided by 3 or 9.
There is no direct relationship between parity (even/odd) and digital root. Even numbers can have any digital root from 1 to 9 (e.g., 10 has digital root 1, 12 has digital root 3, 14 has digital root 5). Similarly, odd numbers can have any digital root. However, if the digital root is even, the number is not necessarily even, and vice versa.
The digital root of a number equals the remainder when the number is divided by 9, except that when the remainder is 0, the digital root is 9. For division by 3, the remainder equals the digital root modulo 3: if digital root is 3, 6, or 9, remainder is 0; if digital root is 1, 4, or 7, remainder is 1; if digital root is 2, 5, or 8, remainder is 2.
Explanation
(i) The chapter shows that digital root depends on the sum of digits, while parity depends only on the last digit. These are independent properties. (ii) The chapter establishes that the digital root is directly related to divisibility by 9. Since 3 divides 9, the remainder when dividing by 3 can be found from the digital root by taking it modulo 3.