Let be the statement ' is divisible by 9' and be the statement 'the sum of the digits of is divisible by 9'. Which of the following is correct? (Here is a positive integer.)
Answer and explanation
Correct answer: D. is both a necessary and a sufficient condition for
- Step 1: A number is congruent to its digit sum modulo , because and so every power of is .
- Step 2: So is divisible by exactly when its digit sum is: the statements are equivalent.
Answer
is both a necessary and a sufficient condition for .
Why not the others: There is no number whose divisibility by differs from that of its digit sum.
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