SOLUTION: Find all numbers between 200 and 500 which have exactly 9 factors.

Algebra.Com
Question 1182879: Find all numbers between 200 and 500 which have exactly 9 factors.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Find all numbers between 200 and 500 which have exactly 9 factors.
~~~~~~~~~~~~~~~~~~ 

Notice that 9 = ,  and it tells us that the south number N is the square 
of the integer positive number  " n ",  which is the product of two prime numbers.


So, we look into the open interval from   ~ 14.1  to   ~ 22.4


and search there the numbers  " n "  that are the products of two different prime numbers.


We find there ONLY THREE such numbers  15, 21 and 22,  so the  ANSWER  to the problem's question 

are the numbers   = 225,   = 441.  and   = 484.

Solved.


--------------


About the relevant theorem from the number theory on the number of divisors to any integer positive number see the lesson

    - Problems on divisors of a given number

in this site.


This theorem states 
    For integer number  N = ,  where p, q, . . . , r are prime divisors  

    and  ,  , . . . ,  are integer exponents  (indexes),  the number of divisors is  .



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


I will assume, since you are working a problem like this, that you know how to find the number of factors of a given number.

To get exactly 9 factors, the number can be either
(a) -- the 8th power of a single prime p, because 8+1=9; or
(b) -- the product of the squares of two primes p and q, because (2+1)(2+1)=9

For case (a), 2^8=256, which is in the required range.

For case (b), we need the number to be the product of two primes; and the number must be between sqrt(200) and sqrt(500) -- or 15 to 22, inclusive. The numbers that satisfy those conditions are 15=3*5, 21=3*7, and 22=2*11.

ANSWER:
256 = 2^8
225 = 15^2
441 = 21^2
484 = 22^2
RELATED QUESTIONS

find the sum of all the numbers between 200 and 600 which are divisible by... (answered by Theo,MathTherapy,math_helper)
Which numbers between 150 and 200 are divisable by... (answered by natroa08,rfer)
How many numbers are there between 500 and 1000 which exactly one their digits as... (answered by Edwin McCravy)
find the sum of all natural numbers between 1 and 100 which are not exactly divisible by... (answered by KMST)
how many numbers are there between 200 and 500 which are multiples of 3 and whose cube... (answered by Alan3354)
How many numbers between 1 and 1000 contain all the numbers 2 3,4,5,6 as... (answered by greenestamps)
what prime numbers are less then 200 but, have the prime factors 2,3,and... (answered by Alan3354)
What is the sum of all natural numbers between 1 and 200 which are multiples of... (answered by htmentor,greenestamps)
how many numbers between 200 and 500 are divisible by... (answered by ikleyn,MathTherapy)