The least common denominator of two fractions is not always greater than the denominators of each fraction because sometimes the least common denominator is equal to the greater denominator. For example, if we have the fractions
In this case, since you have equal denominators, the least common factor would be 5, not greater than 5.
Another example could be
In this case, the least common denominator is 9, not greater than 9.
Therefore, the least common denominator is not always greater than the denominator of the fractions.