If the GCF (x, 36) is 4 and LCM (x, 36) is 288, then find the number x.
LCM(36,x) = 36x / GCD(36,x)
288 = 36x/4
x = 32
32
Well, it looks like x has some commitment issues. It's having a hard time deciding between 4 and 288!
To find the number x, we'll need to use the relationship between the Greatest Common Factor (GCF) and the Least Common Multiple (LCM) of two numbers.
Given that the GCF (x, 36) is 4, it means that 4 is a common factor of both x and 36.
The prime factorization of 36 is 2^2 * 3^2, which means that 36 can be written as 2 * 2 * 3 * 3.
Since 4 is a common factor of x and 36, we can write x as x = 4 * a, where a represents the remaining factors of x.
Now, let's consider the LCM (x, 36) = 288. The LCM is the smallest multiple that two numbers have in common.
Since 288 is a multiple of 36 (2^2 * 3^2), the prime factorization of 288 is 2^5 * 3.
Now, we can find the remaining factors of x. Since x = 4 * a and 288 = 2^5 * 3, the prime factors of x that are not already present in 36 should be: 2^3 (to make a total of 2^4) and 3 (to make a total of 3^1).
Therefore, x = 4 * a = 2^4 * 3^1 = 16 * 3 = 48.
So, the number x is 48.