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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.

ğ‘Ž, 𝑏 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 ↔ (ğ‘Ž + 1 = 𝑏) ∨ (𝑏 + 1 = ğ‘Ž)

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

ğ‘Ž ∈ 𝑍 𝑖𝑠 ğ‘Ž 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 ğ‘ ğ‘žğ‘¢ğ‘Žğ‘Ÿğ‘’ ↔ ∃𝑏 ∈ 𝑍 | ğ‘Ž = 𝑏 2

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