The Numbers on a Dice

 Follow @ProbabilityPuz 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) A Perfect Mathematical Christmas Gift The Best Books to Learn Probability

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