The GCF of 45 and x is 9.
45 is made of 3 * 3 * 5
x is made of 3 * 3 * ? (no 5's)
The LCM of 45 and x is 270.
45 is (still) made of 3 * 3 * 5
270 is made of 2 * 3 * 3 * 3 * 5
For 270, we knock out the parts from the 45 (3 * 3 * 5), leaving us with 2 * 3. That 2 * 3 has to come from the x, so x is 3 * 3 * 2 * 3 = 54.
Check:
Do 45 and 54 give us a GCF of 9?
Divide 45 and 54 by 9 to get 5 and 6.
Are there any common factors in 5 and 6?
No.
45 and 54 have a GCF of 9.
.
Do 45 and 54 give us an LCM of 270?
Prime factors:
45 = 3 * 3 * 5
54 = 2 * 3 * 3 * 3
270 = 2 * 3 * 3 * 3 * 5
2 * 3 * 3 * 3 * 5 is the smallest list of primes that gives us enough primes to make 45 and 54, so 2 * 3 * 3 * 3 * 5 = 270 is the LCM of 45 and 54.
.
Answer: x is 54.