# Question: determine whether the statement is true or false any product...

###### Question details

Determine whether the statement is true or false.

Any product of four consecutive integers is one less than a perfect square.

Justify your answer with a proof or a counterexample, as appropriate. Use only the definitions of the terms and laws of algebra to justify the statements in your proof.

Note:

Two integers are consecutive if, and only if, one is one more than the other.

𝑎, 𝑏 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 ↔ (𝑎 + 1 = 𝑏) ∨ (𝑏 + 1 = 𝑎)

Note: An integer 𝑎 is a perfect square if and only if it equals the square of an integer.

𝑎 ∈ 𝑍 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 ↔ ∃𝑏 ∈ 𝑍 | 𝑎 = 𝑏 ^{2}