Question

If $n = 1 + X$ where $X$ is the product of four consecutive positive integers, which of the following are true?
1.\ $n$ is odd 2.\ $n$ is prime 3.\ $n$ is a perfect square.

• A. 1 and 3 only
• B. 1 and 2 only
• C. 1 only
• D. none of these

Hint:

Memorize the identity: $k(k+1)(k+2)(k+3)+1=(k^2+3k+1)^2$. It instantly shows the result is an odd perfect square (and thus not prime for $k\ge 1$).

Answer 1

The Correct Option is A

Step 1: Represent X using a parameter.
Let the four consecutive integers be k, k+1, k+2, k+3 (with k ∈ ℤ_>0). Then:
X = k(k+1)(k+2)(k+3), n = X + 1.

Step 2: Prove the perfect-square identity.
Observe:
k(k+1)(k+2)(k+3) + 1 = (k² + 3k + 1)².
Hence n is always a perfect square ⇒ Statement 3 is true.

Step 3: Check parity of n.
Among four consecutive integers there are two even numbers, so X is even. Therefore n = X+1 is odd ⇒ Statement 1 is true.

Step 4: Is n prime?
Since n = (k² + 3k + 1)² and k ≥ 1, we have n > 1 and n is a non-trivial square, hence composite. Statement 2 is false.

Answer: Statements 1 and 3 only are true.