If the product of two positive integers is 363, and the least common multiple of them is 33, what is the sum of the two positive integers?
33 = 3x11
and 363/33 = 11
I would say the two numbers are 11 and 33
so the sum of the two numbers is 44
check:
product: 11(33) = 363
LCM of 11 and 33 is 33
Let's solve this problem step by step:
Step 1: Find the prime factorization of 363.
We can start by finding the prime factorization of 363. This means expressing 363 as a product of prime numbers.
The prime factorization of 363 is: 3 x 11 x 11.
Step 2: Find the prime factorization of 33.
Similarly, we find the prime factorization of 33.
The prime factorization of 33 is: 3 x 11.
Step 3: Determine the common factors.
To find the least common multiple (LCM), we need to determine the common factors between the two numbers. In this case, the common factors are 3 and 11. Since these factors contribute to both numbers, they must be in the LCM.
Step 4: Determine the remaining factors.
Next, we need to consider the remaining factors that are unique to each number. For the first number, 363, we have another factor of 11. For the second number, 33, we have another factor of 3.
Step 5: Calculate the LCM.
Now, we can calculate the least common multiple (LCM) by multiplying all the common and remaining factors together:
LCM = common factors x remaining factors
= 3 x 11 x 11 (common factors) x 11 (remaining factor from 363) x 3 (remaining factor from 33)
= 33 x 11 x 11
= 363
Since the LCM is equal to 33, we know that the remaining factors multiply together to give 1. Therefore, there is no additional factor to consider.
Step 6: Calculate the sum of the two positive integers.
The two positive integers are the product of the prime factors: 33 and 11.
Sum = 33 + 11 = 44
Therefore, the sum of the two positive integers is 44.