2018 UNCO Math Contest II Problems/Problem 7

Problem

Let $x = 2^A + 10^B$ where $A$ and $B$ are randomly chosen with replacement from among the positive integers less than or equal to twelve. What is the probability that $x$ is a multiple of $12$ ?

Solution

To have a number divisible by $12$ , it must be divisible by $3$ and $4$ .

Consider mod 3 of x:

$x \: {\equiv} \: 0 \: (mod \: 3)$

$2^A + 10^B\:{\equiv}\:0\:(mod\:3)$

$2^A + 1\:{\equiv}\:0\:(mod\:3)$

$2^A\:{\equiv}\:2\:(mod\:3)$

Here, since $2^A\:{\equiv}\:1 \: or \: 2 \: (mod \: 3)$ for A is even and odd respectively

$\therefore \: A$ is odd

Consider mod 4 of x:

$x \: {\equiv} \: 0 \: (mod \: 4)$

$2^A + 10^B\:{\equiv}\:0\:(mod\:4)$

$2^A + 2^B\:{\equiv}\:0\:(mod\:4)$

$\because\:$ we know A is odd, for $A=1$ , taking $mod\:4$ gives $2$ as result; for $A>1$ , taking $mod\:4$ gives $0$ as result, so we split the case for $A=1$ and $A>1$ here.

For $A=1$ ,

$2 + 2^B\:{\equiv}\:0\:(mod\:4)$

$2^B\:{\equiv}\:2\:(mod\:4)$

$\therefore \: B=1$

For $A>1$ ,

$2^B\:{\equiv}\:0\:(mod\:4)$

$\therefore \: B>1$

$\therefore\:$ Concluding our above conditions, we have $(A,B) = (1,1)$ or $(A,B) \in (\{3,5,7,9,11\},\:\{2,3,4,5,6,7,8,9,10,11,12\})$

By counting the number of solutions, the required probability $=\frac{1\times1 + 5\times11}{12\times12} = \frac{56}{144} = \frac{7}{18}$

2018 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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2018 UNCO Math Contest II Problems/Problem 7

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