Look at each of the following statements. Which are correct and why? (i) If a number is divisible by 9, then the sum of its digits is divisible by 9. (ii) If the sum of the digits of a number is divisible by 9, then the number is divisible by 9. (iii) If a number is not divisible by 9, then the sum of its digits is not divisible by 9. (iv) If the sum of the digits of a number is not divisible by 9, then the number is not divisible by 9.
Correct. This is a direct consequence of the divisibility rule for 9. If a number is divisible by 9, its digit sum must be divisible by 9.
Correct. This is the converse of the divisibility rule. If the digit sum is divisible by 9, the number itself must be divisible by 9.
Correct. Since a number and its digit sum have the same remainder when divided by 9, if a number is not divisible by 9, its digit sum cannot be divisible by 9.
Correct. This is the contrapositive of statement (ii). If the digit sum is not divisible by 9, the number cannot be divisible by 9.
Explanation
(i) The chapter proves that the remainder when a number is divided by 9 equals the remainder when its digit sum is divided by 9. Therefore, if a number is divisible by 9 (remainder 0), its digit sum must also leave remainder 0 when divided by 9, meaning the digit sum is divisible by 9. (ii) The chapter shows that a number and its digit sum have the same remainder when divided by 9. If the digit sum is divisible by 9 (remainder 0), then the number also has remainder 0 when divided by 9, meaning the number is divisible by 9. (iii) The chapter establishes that the remainder when dividing by 9 is the same for a number and its digit sum. If a number is not divisible by 9, it has a non-zero remainder. Therefore, its digit sum must also have a non-zero remainder, meaning the digit sum is not divisible by 9. (iv) The chapter shows that divisibility by 9 and digit sum divisibility by 9 are equivalent conditions. If the digit sum is not divisible by 9 (has a non-zero remainder), then the number also has a non-zero remainder when divided by 9, meaning the number is not divisible by 9.