Question 1182879
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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.<br>
To get exactly 9 factors, the number can be either
(a) {{{p^8}}} -- the 8th power of a single prime p, because 8+1=9; or
(b) {{{(p^2)(q^2)}}} -- the product of the squares of two primes p and q, because (2+1)(2+1)=9<br>
For case (a), 2^8=256, which is in the required range.<br>
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.<br>
ANSWER:
256 = 2^8
225 = 15^2
441 = 21^2
484 = 22^2<br>