Partners in Integers

Two integers are said to be partners if both are divisible by the same set of prime numbers. Find the number of positive integer less than 25 that have no partners less than 25.
Source: NCTM Mathematics Teacher, August 2006

Solution
24=2^3\!\cdot\! 3
18=2\!\cdot\! 3^2
12=2^2\!\cdot\! 3
6=2\!\cdot\! 3
24,18,12,6 are partners; all are divisible by \{2,3\}
20=2^2\!\cdot\! 5
10=2\!\cdot\! 5
20,10 are partners; all are divisible by \{2,5\}
16=2^4
8=2^3
4=2^2
2=2
16,8,4,2 are partners; all are divisible by \{2\}
9=3^2
3=3
9,3 are partners; all are divisible by \{3\}

12 integers have no partners: 1,5,7,11,13,14,15,17,19,21,22,23

Answer: 12

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About mvtrinh

Retired high school math teacher.
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