Difference between revisions of "2004 AMC 10B Problems/Problem 4"

Latest revision as of 22:42, 16 January 2024

Problem

A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$ ?

$\mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 144\qquad \mathrm{(E) \ } 720$

Solution 1

The product of all six numbers is $6!=720$ . The products of numbers that can be visible are $720/1$ , $720/2$ , ..., $720/6$ . The answer to this problem is their greatest common divisor -- which is $720/L$ , where $L$ is the least common multiple of $\{1,2,3,4,5,6\}$ . Clearly $L=60$ and the answer is $720/60=\boxed{\mathrm{(B)}\ 12}$ .

Solution 2

Clearly, $P$ cannot have a prime factor other than $2$ , $3$ and $5$ .

We can not guarantee that the product will be divisible by $5$ , as the number $5$ can end on the bottom.

We can guarantee that the product will be divisible by $3$ (one of $3$ and $6$ will always be visible), but not by $3^2$ .

Finally, there are three even numbers, hence two of them are always visible and thus the product is divisible by $2^2$ . This is the most we can guarantee, as when the $4$ is on the bottom side, the two visible even numbers are $2$ and $6$ , and their product is not divisible by $2^3$ .

Solution 3

The product P can be one of the following six numbers excluding the number that is hidden under, so we have: \begin{align*} 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 2^4 \cdot 3^2 \cdot 5 \\ 1 \cdot 3 \cdot 4 \cdot 5 \cdot 6 = 2^3 \cdot 3^2 \cdot 5 \\ 1 \cdot 2 \cdot 4 \cdot 5 \cdot 6 = 2^4 \cdot 3 \cdot 5 \\ 1 \cdot 2 \cdot 3 \cdot 5 \cdot 6 = 2^2 \cdot 3^2 \cdot 5 \\ 1 \cdot 2 \cdot 3 \cdot 4 \cdot 6 = 2^4 \cdot 3^2 \\ 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 2^3 \cdot 3 \cdot 5 \end{align*}

The largest number that is certain to divide product P is basically GCD of all the above 6 products which is $2^2 \cdot 3$ .

Hence $P=3\cdot2^2=\boxed{\mathrm{(B)}\ 12}$ .

@@ Line 32: / Line 32: @@
 \end{align*}
-The largest number that is certain to divide product P is basically gcd of all the above 6 products which is <math>2^2 \cdot 3.
+The largest number that is certain to divide product P is basically GCD of all the above 6 products which is <math>2^2 \cdot 3</math>.
-Hence </math>P=3\cdot2^2=\boxed{\mathrm{(B)}\ 12}$.
+Hence <math>P=3\cdot2^2=\boxed{\mathrm{(B)}\ 12}</math>.
 == See also ==

2004 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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