The relationship between two positive integers a and b is defined as the ratio of the least common multiple of a and b to the product of a and b. For example, the least common multiple of 10 and 15 is 30, while the product of 10 and 15 is 150, resulting in a relationship of 30/150 = 1/5. A positive integer y is less than 20, and the relationship between y and 10 equals 1/1. How many possible values of y exist?

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Official Explanation

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The relationship between y and 10 being 1/1 implies that LCM(10, y) = 10y. This indicates that 10 and y are co-prime, meaning they share no common factors other than 1. To find how many integers less than 20 are co-prime with 10, we need to determine how many integers less than 20 are not divisible by 2 or 5 (since 10 = 2 * 5). There are (18-2)/2 + 1 = 9 multiples of 2 in the range from 0 to 20, not inclusive; There are (18-5)/5 + 1 = 4…

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What this question tests

What it tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Common trap

Forgetting that 1 is not prime and that every integer divides 0.