A common denominator is 75.
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The common denominator of any two or more whole numbers will always be 1.
To find a fraction's simplest form, you first need to identify any common divisors between the numerator and the denominator. In this case the numerator (27) and the denominator (75) can both divide by 3. If we do this we get 9/25. There are no more common factors. Thus the simplest form of 27/75 is 9/25.
LCD(75, 275) = 825
A common denominator, though not the least, is 150.
If I wanted to be nasty about it, I would say 1, because the least common denominator of 45/1 and 75/1 is 1. The least common denominator of two fractions is the least common multiple of their denominators. I presume you are asking for the least common multiple of 45 and 75, which is 225.