Highly Composite Number Calculator

Input a number to determine if it is highly composite, or input a range to generate all highly composite numbers within it.

Highly Composite Number Check or Generate

What Is a Highly Composite Number?

A highly composite number is a positive integer that has more divisors than any smaller positive integer. In other words, if a number has more factors than any number less than it, it is classified as highly composite.

How to Determine if a Number Is Highly Composite

  1. Find all the divisors of the number and count them.
  2. Compare this count with the divisor counts of all integers less than the given number.
  3. If no smaller number has more divisors, the number is highly composite; otherwise, it is not.

Examples

Example 1: Is 12 a Highly Composite Number?

Solution:

Divisors of 12: 1, 2, 3, 4, 6, 12.

Total divisors: 6

Compare divisor counts for numbers 1 to 11:

1: 1

2: 2

3: 2

4: 3

5: 2

6: 4

7: 2

8: 4

9: 3

10: 4

11: 2

The number 12 has the most divisors (6).

Result: 12 is a highly composite number.

Example 2: Is 24 a Highly Composite Number?

Solution:

Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Total divisors: 8

Compare divisor counts for numbers 1 to 23:

The highest divisor count is 6 (from 12).

8 > 6

Result: 24 is a highly composite number.

Example 3: Is 144 a Highly Composite Number?

Solution:

Divisors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144

Total divisors: 15

Compare divisor counts for numbers 1 to 143:

The highest divisor count is 16 (from 120).

15 < 16

Result: 144 is not a highly composite number.