The gcf of two numbers is 3 and third lcm is 180 if one of numbers is 45 then find the second number
Let's assume the second number is x.
We know that the GCF of two numbers is 3. Therefore, both 45 and x must be divisible by 3.
Since the GCF of two numbers is the largest common divisor, and we are given that one of the numbers is 45, it means that 45 is the GCF. Thus, 45 and x are both multiples of 45.
We are also given that the LCM of the two numbers is 180. The LCM is the smallest common multiple, so 180 must be divisible by both 45 and x.
Let's express 180 as a product of its prime factors:
180 = 2^2 * 3^2 * 5
Since 180 must be divisible by 45, it means that 180 must contain all the prime factors of 45. In other words, 180 must contain at least two 3s and one 5.
Let's check if 180 is divisible by x:
180 = 2^2 * 3^2 * 5
This expression already contains two 3s and one 5. Therefore, no matter what the value of x is, 180 is divisible by x.
So, the second number (x) can be any factor of 180 that is divisible by 3.
The factors of 180 that are divisible by 3 are: 3, 6, 9, 15, 18, 30, 45, 60, 90, and 180.
However, we know that x has to be the GCF, which is 45. Therefore, the second number is 45.