Let be the statement ' is divisible by 3' and be the statement ' leaves remainder 1 when divided by 3'. Which of the following is correct? (Here is a positive integer.)

Answer and explanation

Correct answer: B. is both a necessary and a sufficient condition for

  1. Step 1: Check the three possible remainders of modulo : gives ; gives ; gives .
  2. Step 2: So is divisible by exactly when : the statements are equivalent.
Answer
is both a necessary and a sufficient condition for .
Why not the others: There are only three cases to check, and exactly one of them makes the expression a multiple of .

Practise more Necessary and sufficient conditions.

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