Prime Factor Subsets
For any highly composite number, if one takes any subset of prime factors for that number and their exponents, the resulting number will have more divisors than any smaller number that uses the same prime factors. For example for the highly composite number 720 which is 24 × 32 × 5 we can be sure that
- 144 which is 24 × 32 has more divisors than any smaller number that has only the prime factors 2 and 3
- 80 which is 24 × 5 has more divisors than any smaller number that has only the prime factors 2 and 5
- 45 which is 32 × 5 has more divisors than any smaller number that has only the prime factors 3 and 5
If this were untrue for any particular highly composite number and subset of prime factors, we could exchange that subset of primefactors and exponents for the smaller number using the same primefactors and get a smaller number with at least as many divisors.
This property is useful for finding highly composite numbers.
Read more about this topic: Highly Composite Number
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