Page 87 - 'Blast_Into_Math

Blast into Math! The Euclidean algorithm: a computational recipe

For n and k ∈ N , simplify

(n + k)!
.
n!

10. *Give an example of four positive integers such that any three of them have a common
divisor greater than 1 although only 1 and −1 divide all four of them.

4.6 Examples and hints

In mathematics research, we solve mathematical problems which have never been solved. So, when we
get stuck, what do we do? How do we “ask for help” when no one knows the answer to our problem?
First, we can talk to another mathematician, which is like talking to your classmate. Even if the other
mathematician can’t solve our problem, he or she could have new ideas about it or see it from a different
perspective. Next, we can read articles or books about similar and related problems, which is like studying
examples or reading your textbook. However, unlike in a class where the teacher selects the textbook,
we must find articles and books ourselves. This is one of the reasons there are problems in this book

which ask you to find information on your own. Finally, if we have worked several consecutive hours
on the same problem, it may be time to take a break. Get some fresh air! Do some physical activities,
play some music; whatever it may be, do something completely different from mathematics. When you
come back to the problem later, you’ll be able to face it with new mathematical energy!

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