Hello from MrBillDoesMath!
Answer:
One example is 52 and 54. They are between 50 and 60 and have the factor 2 in common
Discussion:
Consider the numbers between 50 and 60:
51, 52, 53, 54, 55, 56, 57, 58, 59
53 and 59 are primes so they don't have any factors in common and we can focus on the other numbers in the range.
Consider 52. It's even so is divisible by 2 as are 54, 56, 58. So 52 and 54 , for example, are are two number between 50 and 60 that have the factor 2 in common
Even number are visible by 2. Let's look for numbers visible by 3. How about 54 ( = 18 *3) and 57 ( = 19*3)? That works so its looks like there are many answers to this question.
Thank you,
MrB
