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Printjmc
counting and probability senior
Problem
The integers and are randomly selected, where and . What is the probability that the division is an integer value? Express your answer as a common fraction.
Solution
The possible values of are represented by the set and for the set There are thus pairs of integers.
Now, we see which satisfy the divisibility requirement that . If then can only be 2, or 1 integer. If , then can be no integer. If , then can be any integer, or 6 choices. If , then cannot be any integer. If , then can only be 2, or 1 integer. If then can only be 3, or 1 integer. If , then can be 2 or 4, or 2 different integers. If , then is the only possibility, for 1 integer. So, possibilities. So, is the probability of being an integer.
Now, we see which satisfy the divisibility requirement that . If then can only be 2, or 1 integer. If , then can be no integer. If , then can be any integer, or 6 choices. If , then cannot be any integer. If , then can only be 2, or 1 integer. If then can only be 3, or 1 integer. If , then can be 2 or 4, or 2 different integers. If , then is the only possibility, for 1 integer. So, possibilities. So, is the probability of being an integer.
Final answer
\frac{1}{4}