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

i know 2^8 is an option

catswift Sep 18, 2024

#1**0 **

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.

tomtom Sep 18, 2024