among 3 digit numbers, how many numbers are there with exactly 9 factors? help

Here's how we can find the number of 3-digit numbers with exactly 9 factors:

 

Factorization: A number with 9 factors must be a perfect square of a number with 5 factors (since the factors come in pairs). For example, 256 has 9 factors because it's the square of 16, which has 5 factors (1, 2, 4, 8, 16).

 

Perfect Squares: To find 3-digit perfect squares, we can start by listing the perfect squares from 100 to 999: 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900.

 

Counting Factors: Now, we need to check which of these perfect squares have a number with 5 factors. We can do this by prime factorizing each number and counting the number of factors using the formula: (exponent of factor 1 + 1) * (exponent of factor 2 + 1) * ...

 

Here's a table showing the prime factorization and number of factors for each perfect square:

 

Perfect SquarePrime FactorizationNumber of Factors

 

100 2^2 * 5^29

 

121 11^23

 

144 2^4 * 3^225

 

225 3^2 * 5^29

 

256 2^89

 

324 2^2 * 3^415

 

400 2^4 * 5^225

 

441 3^2 * 7^29

 

484 2^2 * 11^29

 

529 23^23

 

676 2^2 * 13^29

 

841 29^23

 

900 2^2 * 3^2 * 5^227

 

Counting Numbers with 9 Factors: Finally, we count the number of perfect squares in the table that have exactly 9 factors. There are four such numbers: 100, 256, 441, and 841.

 

Therefore, there are four 3-digit numbers with exactly 9 factors.