To find the value of 2X – 3Y, we need to first determine the values of X and Y. Here's how we can solve the problem step by step:

Step 1: Prime factorization of 77
To find the prime factorization of 77, we divide it by the smallest prime number, which is 2. However, 2 does not divide 77. Next, we try dividing it by the next prime number, which is 3. We find that 3 divides 77, resulting in 77 ÷ 3 = 25 remainder 2. Continuing with the prime factorization, we divide 25 by the next prime number, which is 5. We find that 5 divides 25, resulting in 25 ÷ 5 = 5. Therefore, the prime factorization of 77 is 7 × 11.

Step 2: Possible values of X and Y
Since the least common multiple (LCM) of X and Y is 77, both X and Y must have the prime factors of 7 and 11. However, since X is greater than Y, X must have at least one additional prime factor. Therefore, the possible values of X and Y are as follows:
- X = 7 × 11 × P (where P is a prime number greater than 11)
- Y = 7 × 11

Step 3: Finding the value of 2X – 3Y
Substituting the values of X and Y into the expression 2X – 3Y, we get:
2(7 × 11 × P) – 3(7 × 11)
= 154P – 231
= 7(22P – 33)

Step 4: Simplifying the expression
We can simplify the expression further by factoring out the common factor of 7:
7(22P – 33)
= 7(11 × 2P – 11 × 3)
= 7(11(2P – 3))

Step 5: Determining the value of the expression
Since both X and Y are prime numbers, P must also be a prime number. Therefore, the expression 2X – 3Y will only be a multiple of 7 if the expression inside the parentheses is divisible by 7. This occurs when 2P – 3 is a multiple of 7. The only prime number P that satisfies this condition is P = 5.

Substituting P = 5 into the expression, we get:
2X – 3Y = 7(11(2P – 3))
= 7(11(2(5) – 3))
= 7(11(10 – 3))
= 7(11(7))
= 7(77)
= 539

Therefore, the value of 2X – 3Y is 539.