if any nonsquare number has 18 different rectangular arrays, how many factors does it havw

Answer 1

If a non-square number has 18 different rectangular arrays, it means it has 18 different pairs of factors.

The number of factors of a number can be calculated by finding the prime factorization of the number and adding 1 to each exponent, and then multiplying the results. For example, the factors of 18 can be found by expressing it as a product of prime numbers: 18 = 2^1 * 3^2. Adding 1 to each exponent gives (1+1) * (2+1) = 2 * 3 = 6. Therefore, 18 has 6 factors.

So if a non-square number has 18 different rectangular arrays, it has 18 pairs of factors, and the total number of factors can be calculated in a similar way. We need to find two numbers whose exponents multiply to 18, and then add 1 to each exponent and multiply the results.

The prime factorization of 18 is 2^1 * 3^2. To find pairs of exponents that multiply to 18, we can consider the following possibilities:

1 * 18 = (1+1) * (18+1) = 2 * 19 = 38 factors
2 * 9 = (2+1) * (9+1) = 3 * 10 = 30 factors
3 * 6 = (3+1) * (6+1) = 4 * 7 = 28 factors

So a non-square number with 18 different rectangular arrays can have either 38, 30, or 28 factors, depending on the specific combinations of exponents in its prime factorization.