2018 UNCO Math Contest II Problems/Problem 7
Let where and are randomly chosen with replacement from among the positive integers less than or equal to twelve. What is the probability that is a multiple of ?
To have a number divisible by , it must be divisible by and .
Consider mod 3 of x:
Here, since for A is even and odd respectively
Consider mod 4 of x:
we know A is odd, for , taking gives as result; for , taking gives as result, so we split the case for and here.
Concluding our above conditions, we have or
By counting the number of solutions, the required probability
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