May 19th
May 19th
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
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.
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