Let G be the GCD of three numbers 3,780, 3,465, and 3,003 and L be the LCM of 3,465 and 3,003. Then \(\frac{L}{33G}\) = 

Given:

\(G = \text{GCD of } 3780,\ 3465,\ 3003\)

\(L = \text{LCM of } 3465,\ 3003\)

Find: \(\frac{L}{33G}\)

Formula used:

GCD = Common HCF of given numbers

LCM = \(\frac{a \times b}{\text{GCD}(a, b)}\)

Calculations:

Prime factorization:

3780 = 2 × 2 × 3 × 3 × 3 × 5 × 7 = 2² × 3³ × 5 × 7

3465 = 3 × 3 × 5 × 7 × 11 = 3² × 5 × 7 × 11

3003 = 3 × 7 × 11 × 13 = 3 × 7 × 11 × 13

⇒ G = Common factors = 3 × 7 = 21

LCM of 3465 and 3003 = \(\frac{3465 × 3003}{\text{GCD}(3465, 3003)}\)

GCD(3465, 3003):

3465 = 3² × 5 × 7 × 11

3003 = 3 × 7 × 11 × 13

⇒ GCD = 3 × 7 × 11 = 231

⇒ LCM = (3465 × 3003) ÷ 231

⇒ LCM = 10399995 ÷ 231 = 45045

Now, \(\frac{L}{33G} = \frac{45045}{33 × 21} = \frac{45045}{693}\)

⇒ \(\frac{45045}{693} = 65\)

∴ The correct answer is 65.