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Fields Medal Prize Winners (1998)

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Positive Integral Divisors

1 In order to understand positive integral divisors, we need to understand divisors, prime numbers and prime factorizations, and how each of these parts of number theory work with each other.

 

Divisors, Primes, and Prime Factorization

If a whole number is divided by a whole number and the quotient is a whole number, then the two numbers are called divisors of the originals number. For example, since 24 ÷ 6 = 4 and all the numbers are whole numbers, we can say that 6 is a divisor of 24. Also notice that 4 is a divisor of 24 too, since we can change the order to get 24 ÷ 4 = 6. Therefore, both 4 and 6 are divisors of 24. Other divisors of 24 include 1, 2, 3, 8, 12, and 24.

A prime number is a positive number whose only divisors are 1 and itself.

The prime factorization of a number is written as the product of the prime numbers that make up that number.

 

Example:

Find the prime factorization of 24.

The prime factorization of 24 is 2 × 2 × 2 × 3, since these are the primes that need to be multiplied to make up 24. This number can also be written in exponent form as 2 3 × 3, since it is the product of three 2's and one 3,

 

Example:

Find the prime factorization of 64. The prime factorization of 64 is 2 × 2 × 2 × 2 × 2 × 2 = 2 6 .

 

Find the Number of Positive Integral Divisors

To find how many positive integral divisors a number has, there are two methods to choose from. In the first method, you just count the divisors mentally. We know that each number has positive integral divisors in pairs, except for perfect square numbers. If you know your number is not a perfect square, then you can count in pairs.

 

Example:

How many positive integral divisors does 24 have?

You can quickly run through the numbers, starting with 1, to find the positive integral divisors of 24. Considering pairs, we start with 1 and 24. Both of those numbers are positive integral divisors and our count is up to 2. Next is 2 and 12. The count is up to 4. After that, 3 and 8 are positive integral divisors and the count is now 6. Finally, we have 4 and 6. The count is 8. We consider 5, but 24 ÷ 5 is not an integer (whole number), so it is not a positive integral divisor. The next number is 6, but we have already accounted for 6 with the 4. When you reach a number that you have already accounted for, you stop counting. Therefore, there are 8 positive integral divisors for 24. 24

24
1 24
2 12
3 8
4 6

 

Advanced Idea: Is there a way to know when to stop counting and that we have reached the last number before we start repeating numbers? Yes, there is! We know that for every positive number has a square root. The square root of a number is the number you can multiply by itself to get the original number. For example, the square root of 49 is 7 because 7 × 7 = 49. What about numbers that are not perfect squares? In this case we just need to know approximately where the square root is. The square root of 24 is approximately 4.9. Therefore, any number we try to count AFTER 4.9 is going to be a repeat from our list. Remember that 4 was the last number we tried and it was a positive integral divisor of 24, 5 was not and 6 was a repeat (from the 4). Knowing that 4.9 was the square root of 24 and thus, the limit of our search, we could have stopped at 4 and not bothered with 5 or 6. This trick can save some time.

This previous method works fairly well with small numbers. The next method works will all numbers, but requires that you know or can quickly compute a number's prime factorization in exponent notation.

To find the number of positive integral divisors of a number, use the following steps:

  • Compute the numbers prime factorization in exponent notation.
  • Add 1 to each exponent.
  • Find the product of these new numbers.

 

Example:

Find the number of positive integral divisors of 24.

  • Prime factorization: 24 = 2 3 × 3 1 . We put 3 1 with the one because we need to work with the exponents.
  • Add 1 to each exponent: 3 + 1 = 4 and 1 + 1 = 2.
  • Find the product of these new numbers: 4 × 2 = 8. There are still 8 positive integral divisors for 24.