Which of the following divisibility or primality claims holds for every positive integer ?

Answer and explanation

Correct answer: B. is divisible by

  1. Step 1: is divisible by and (consecutive integers ), and by (check : for one of is a multiple of ; for , ). So it is divisible by .
  2. Step 2: Counterexamples: gives (prime) but gives ; gives , not divisible by ; gives , not divisible by ; gives .
Answer
is divisible by .
Why not the others: A single value breaks each of the other four statements; only the first holds for all .

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