Suppose the number 1155 is written as the product of three positive integers. What is the smallest possible value of the sum of those three integers?
The prime factorization of 1155 is 3 * 5 * 7 * 11. To find the three positive integers that multiply to 1155 with the smallest possible sum, we want to divide the factors into groups as close to each other in value as possible.
One way to do this is to group the factors as follows:
3 * 5 = 15
7 * 11 = 77
15 * 77 = 1155
The sum of these three integers is 15 + 77 + 1155 = 1247.
Therefore, the smallest possible value of the sum of three positive integers that multiply to 1155 is 1247.
How many different primes are in the prime factorization of 87 x 89 x 91 x 93?
To find the prime factorization of 87 x 89 x 91 x 93, we first need to break down each number into its prime factors.
87 = 3 * 29
89 is a prime number
91 = 7 * 13
93 = 3 * 31
So, the prime factorization of 87 x 89 x 91 x 93 is 3^2 * 7 * 13 * 29 * 31 * 89.
There are 6 different prime numbers in the prime factorization of 87 x 89 x 91 x 93: 3, 7, 13, 29, 31, and 89.