Final answer:
The number 7200 has 54 factors, and 8 of these factors are multiples of 40. The sum of these factors is 90.
Step-by-step explanation:
To find the number of factors of a number, we need to find its prime factorization and then use the exponents to calculate the number of factors. In the case of 7200, the prime factorization is 2^5 * 3^2 * 5^2. The exponents are 5, 2, and 2. To calculate the number of factors, we add 1 to each exponent and multiply them together: (5+1) * (2+1) * (2+1) = 6 * 3 * 3 = 54. So, the number 7200 has 54 factors.
Now, to find the factors that are multiples of 40, we need to consider the prime factors of 40. The prime factorization of 40 is 2^3 * 5. We can find all the factors that are multiples of 40 by combining the powers of these prime factors. In this case, we have 2^3 * 5^1, and to calculate the number of factors, we add 1 to each exponent and multiply them together: (3+1) * (1+1) = 4 * 2 = 8. So, there are 8 factors of 7200 that are multiples of 40.
To find the sum of these factors, we can use the formula for the sum of factors. For a number with prime factorization p1^a1 * p2^a2 * ... * pn^an, the sum of factors is given by (p1^(a1+1) - 1)/(p1 - 1) * (p2^(a2+1) - 1)/(p2 - 1) * ... * (pn^(an+1) - 1)/(pn - 1). In this case, we have 2^3 * 5^1, so the sum of factors is (2^(3+1)-1)/(2-1) * (5^(1+1)-1)/(5-1) = 15 * 6 = 90.