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Short answer: There is none.

There are not common factors of a single number because there cannot be any form of common factor without two or more numbers to compare. Common factors are factors that the numbers being compared have in common. Thus, there cannot be a single number that has the greatest number of common factors.

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Q: Which number less than 100 has the greatest number of common factors?

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Lots of numbers are NOT common factors. For example, 5, 7, 100...Get the greatest common factor of both numbers. All common factors are factors of this greatest common factor. Note that all of them must needs be less than or equal to the greatest common factor. Any other number is NOT a common factor.

48

12

The three greatest whole numbers less than 200 that each have an odd number of factors are 144, 169 and 196.

60, 72, 84, 90 and 96 each have twelve factors.

If we are trying to find the two numbers less than 50 with the greatest common factor, we need to pick two numbers with a larger difference between them, since the greatest common factor between two numbers cannot be greatest than the difference between the two numbers. To create a large difference, we will want one number to be close in value to 50. And, for the greatest common factor, we want the other number itself to be the greatest common factor. The greatest common factor that the larger number can have (since it is larger and thus cannot be the greatest common factor itself) is the number which is half its value. So, if we choose the even number closest in value to 50, we get 48. Then, if we take half of it, we get 24. The greatest common factor of 24 and 48 is 24. This is the largest possible greatest common factor of a pair of numbers less than 50.

60, 72 and 84 are tied with twelve.

60, 72, 84, 90 and 96 each have twelve factors.

The greatest possible 'length' comes from the number with the greatest number of prime factors. The greatest number of factors is created by using the smallest prime number, 2, as a factor as many times as possible. Since 2^9=512 and 2^10=1024, the greatest possible 'length' of a positive integer less than 1000 is 9.

greatest number less then 100 = 99

60, 72, 84, 90 and 96 are tied with 12.

The greatest composite number that is less than 100 is 99. Its factors are 1, 3, 9, 11, 33, and 99. It can't be prime because it has factors other than 1 and itself.

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