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

225, 256, 441, 484

Step-by-step explanation:

Only square numbers can have an odd number of factors, e.g. 4: 1, 2, 4

Therefore only square numbers can have 9 factors.

There are only 8 square numbers between 200 and 500: 225, 256, 289, 324, 361, 400, 441, 484.

One way is to list all the factors of each number and you will find that only 225, 256, 441 and 484 have 9 factors.

The other more efficient way is to acknowledge the fact that if a square number (n x n) has 9 factors, n must have only 4 factors. For example, 225 has 9 factors because it is equal to 15 squared and 15 only has 4 factors: 1, 3, 5, 15.

This way, we can find out that 15, 21 and 22 have exactly 9 factors. However, 256 has 9 factors as well as it is equivalent to 2 to the power of 8. This means that it already has 8 factors: 2, 4, 8, 16...,128, 156. However, we must include 1 as a factor which means it has 9 factors in total as well.