Bill Wagenborg
Problem of The Week #4
Prime Numbers and Divisors
Problem
For this problem we had to find out as much as possible about how many divisors a
number has . We would have to come up with a method to find how many divisors a number has
without listing them out.
Process
I started my work by listing all of the numbers from 1 to 100 and figuring out how many
divisors each had. I then put them into columns with the number of divisors as a heading. The
purpose of doing this was to look at the numbers in order to find some pattern of why each had
that particular number of divisors. I looked at if they shared common multiples that others didn’t,
the sum/product/difference/quotient of the prime numbers before, the first/last digit of each
number in the column, if they were a square number or how close they were to a square number.
All of these were dead ends and all week I could not come up with any solution. I read the
problem again and again and the words prime number kept sticking out. I knew it had something
to do with how I would find my answer, but I just didn’t know what. I then just focused on one
number at a time. I looked at 100.I knew it had nine divisors and I knew the way to get the
number nine had something to do with prime numbers. Then I thought about something I taught
this year that had the word prime in it: prime factoization. I took the prime factorization of 100
and got 2 2 *5 2 . I spent more time playing with these numbers and exponents until I found a way
to get nine. I then tried it with other numbers and it worked too.
Solution
In order to find the number of divisors a number has: put it into prime factorization ,take
each of the exponents in the prime factorization and add one. Finally multiply these two numbers
together. In the case of 100, 2 2 * 5 2 , 2+1 and 2+1, 3 * 3 = 9. 100 has 9 divisors. In the examples
of 1,000, 000 and 1 ,000, 000, 000 the same method can be used. For 1,000,000:26 5 6 , 6+1 and
6+1, 7*7 =49 It has 49 divisors. 1,000,000,000: 2 9 * 5 9 , 9 + 1 and 9 +1, 10 * 10 = 100. It has
100 divisors.
As for coming up with the smallest number with a certain number of divisors, you can
just work in th opposite direction. Take the example of the smallest number with 20 divisors.
Using the above method you would have to add a one to two exponents, multiply them and then
get 20. Since we are looking for a small number we will use the two smallest numbers in
combination to get 20: 4 and 5. If I work in reverse and subtract a one from each I will get the
exponents 3 and 4. The lowest prime numbers(excluding one) are and 2 and 3, so I will put the
exponents with two and three as their bases: 2 4 and3 3 or 23 and 34. The first set gives me an
answer of 432 and the second of 648. So my first set gives me the lowest number: 432.
In conclusion, finding the prime factorization of a number is the key to finding the
number of divisors a number has and for finding other details involving divisors.