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Hello, it is really SIMPLE statement, and you do not need to use heavy artillery as that referred theorem.
This statement is for those BEGINNERS who solve their first problems on divisibility numbers
and make their first steps in learning this subject.
Trace attentively my reasons below.
Of any three consecutive integer numbers, at least one is an EVEN number (divisible by 2)
and exactly one is divisible by 3.
These are self-evident facts.
Hence, the product of any three consecutive integer numbers is divisible by 2*3 = 6.
Solved, proved, explained and completed.
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For your education, look into the lesson
- Introductory problems on divisibility of integer numbers
in this site.
It is written specially for those BEGINNERS like you . . .
By the way, at the end of that lesson, you will find numerous links to other lessons on divisibility numbers.
So, you have this happy opportunity to read about the subject in English.
It is really a fascinating reading (!)