Question

If \( a, b, c \) are all positive integers, with \( 4a>b \), then which of the following conditions is BOTH NECESSARY AND SUFFICIENT for the expression \[ \sqrt{(3)^a (21)^{3a-b} (49)^{2b+c}} \] to be a positive integer?

• A. \( a - b = c \)
• B. \( a - b + 2c \) is divisible by 3
• C. \( a, b, \) and \( c \) are divisible by 3
• D. \( a - b \) and \( c \) are divisible by 3
• E. None of the other conditions is both necessary and sufficient

Hint:

When working with powers, ensure that the exponents satisfy the conditions for the expression to be a perfect square, such as even exponents for all primes.