if any nonsquare number has 18 different rectangular arrays, how many factors does it havw
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.
To find the number of factors that a non-square number has given that it has 18 different rectangular arrays, you can follow these steps:
Step 1: Understand the problem
We are given that a non-square number has 18 different rectangular arrays. We need to determine how many factors this number has.
Step 2: Recall the relationship between factors and rectangular arrays
We know that a rectangular array corresponds to the factors of a number. For a rectangular array with dimensions m x n, the corresponding factor is (m + 1) × (n + 1). So, if we find all the different rectangular arrays, we can calculate how many factors the number has.
Step 3: Find the different rectangular arrays
Since we are given that there are 18 different rectangular arrays, we need to list out all the possible combinations of m and n that satisfy the equation (m + 1) × (n + 1) = 18.
The pairs of m and n are as follows:
(0, 17)
(1, 8)
(2, 5)
(3, 3)
(5, 2)
(8, 1)
(17, 0)
Step 4: Calculate the number of factors
Counting the pairs above, there are seven different rectangular arrays. Therefore, the number of factors for the given non-square number is 7.
So, the answer is 7.
To find the number of factors of a non-square number that has 18 different rectangular arrays, we need to consider the prime factorization of the number.
Let's assume the number is represented by N and its prime factorization is given by N = (p1^a1)(p2^a2)(p3^a3)...(pn^an), where p1, p2, p3,..., pn are prime numbers and a1, a2, a3,..., an are their respective powers.
We are given that N has 18 different rectangular arrays. Each rectangular array is formed by multiplying two factors of N together. Since there are 18 different arrays, it means there are 18 different pairs of factors.
Now, we can calculate the total number of factors of N using the formula:
Number of factors = (a1 + 1)(a2 + 1)(a3 + 1)...(an + 1)
In this case, we have 18 different pairs of factors, which means there are a total of 36 factors (2 factors per pair). So, we can rewrite the equation as:
36 = (a1 + 1)(a2 + 1)(a3 + 1)...(an + 1)
To find the number of factors, we need to factorize 36 into its prime factors:
36 = 2^2 * 3^2
Now we can distribute the prime factors to the corresponding powers of the prime factors of N:
(a1 + 1) = 2^2
(a2 + 1) = 3^2
From here, we can solve the equations to find the values of a1 and a2:
(a1 + 1) = 2^2
(a1 + 1) = 4
a1 = 4 - 1
a1 = 3
(a2 + 1) = 3^2
(a2 + 1) = 9
a2 = 9 - 1
a2 = 8
Therefore, the prime factorization of N = (p1^a1)(p2^a2)(p3^a3)...(pn^an) can be written as:
N = (p1^3)(p2^8)
Now, we can calculate the number of factors by using the formula:
Number of factors = (a1 + 1)(a2 + 1)(a3 + 1)...(an + 1)
Number of factors = (3 + 1)(8 + 1)
Number of factors = 4 * 9
Number of factors = 36
Hence, the non-square number that has 18 different rectangular arrays has a total of 36 factors.