Final answer:
The greatest common divisor (gcd) of the expressions (21n+4) and (14n+3) is 1. None of the provided options are correct because the coefficients of n in both expressions are not multiples of each other, and the constant terms 4 and 3 are coprime.
Step-by-step explanation:
The question is asking for the greatest common divisor (gcd) of two linear expressions, (21n+4) and (14n+3). To find the gcd of these kinds of expressions, the Euclidean algorithm can be used, which involves performing repeated division until a remainder of zero is obtained. However, without going through all the calculations, we can observe that since the coefficients of n in both expressions are not multiples of each other, the gcd cannot depend on n. Instead, it has to be a constant value. By looking at the constant terms (4 and 3), we can see that the gcd cannot be greater than 1 because 4 and 3 are coprime (they have no common divisors other than 1).
Option (a) (7n+1) depends on n and has coefficients that are not divisors of the coefficients in the given expressions. Likewise, options (b) (2n+1), (c) (5n+2), and (d) (3n+4) also depend on n and have coefficients that are inconsistent with the divisors of the coefficients in (21n+4) and (14n+3). Consequently, none of the provided options are correct, and the greatest common divisor is actually 1.