Formalize and prove that the product of two consecutive natural numbers is even.
Formalize and prove that the product of three consecutive natural numbers is divisible by 2.
Formalize and prove that the product of three consecutive natural numbers is divisible by 3.
Formalize and prove that the product of three consecutive natural numbers is divisible by 6.
Formalize and prove that the product of four consecutive natural numbers is divisible by 2.
Formalize and prove that the product of four consecutive natural numbers is divisible by 3.
Formalize and prove that the product of four consecutive natural numbers is divisible by 4.
Formalize and prove that the product of four consecutive natural numbers is divisible by 6.
Formalize and prove that the product of four consecutive natural numbers is divisible by 8.
Formalize and prove that the product of four consecutive natural numbers is divisible by 12.
Formalize and prove that the product of four consecutive natural numbers is divisible by 24.
Formalize and prove that the product of five consecutive natural numbers is divisible by 120.
Alfrothul: Wow.
Dana: Right.
Alfrothul: We have seen these numbers before.
Dana: You mean 2, 6, 24, and 120?
Alfrothul: Yes, the biggest of the divisors.
Bong-sun: Which begs the question.
Anton: The question?
Bong-sun: Well, if all these products are divisible, what is the result of the divisions?
Loki: Need a hand?
Anton (prudently): Er, yes?
Loki: Let me add one more exercise.
Formalize and prove that the product of six consecutive natural numbers is divisible by 720.
Perusing all the exercises above (without solving them, unless of course you absolutely need to) begs the following question: since the product of consecutive numbers is divisible by such other numbers, the result of these divisions by the largest possible such other number is a natural number. What is this natural number?
Thanks are due to Yves Bertot for the initial simple problem at this stage of the lecture notes.
Created [22 Feb 2024]