The Numbers on a Dice

Q: A die is rolled and the numbers on all the visible faces is multiplied together $$=S$$. How would you choose a highest number such that it is guaranteed to divide the product.

A: The puzzle rests on a simple fact of prime factorization. Every number can be expressed as a product of prime numbers. For the given example, assume the face that is not visible is 1. The product of the remainder of numbers is $$6 \times 5\times\ldots \times 2$$ or equivalently $$6!$$. If the face that is invisible is 2, the number $$S$$ is $$\frac{6!}{2}$$, if it is 3 the number $$S$$ is $$\frac{6!}{3}$$ and so on. The number $$6!$$ can be expressed as prime numbers as $$2^{4}\times3^{2}\times 5$$. Let us enumerate the cases for each of the faces that is invisible, the product $$S$$ expressed in terms of prime numbers are as follows
1. $$2^{4}\times3^{2}\times5$$
2. $$2^{3}\times3^{2}\times5$$
3. $$2^{4}\times3^{1}\times5$$
4. $$2^{2}\times3^{2}\times5$$
5. $$2^{4}\times3^{2}$$
6. $$2^{3}\times3^{1}\times5$$
Notice, the solution to our problem is the lowest exponent of each of the prime factors. This yields $$2^{2}\times3^{1}\times5^{0} = 12$$.

Three good books to own to learn number theory,

Number Theory: A Lively Introduction with Proofs, Applications, and Stories

An Introduction to the Theory of Numbers

Elementary Number Theory (Springer Undergraduate Mathematics Series)

The Best Books to Learn Probability

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own Discovering Statistics Using R This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing. Fifty Cha

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