Necessary and Sufficient Condition for Integer Value

Rewrite the expression under the cube root in terms of prime factors 3 and 7: $3^a \cdot (3 \cdot 7)^{3a-b} \cdot (7^2)^{2b+c} = 3^{a + 3a - b} \cdot 7^{3a - b + 2(2b + c)} = 3^{4a - b} \cdot 7^{3a + 3b + 2c}$. Taking the cube root, the powers must be multiples of 3 for the expression to be an integer:

1. $4a - b = 3a + (a - b)$ must be divisible by 3, which requires $(a - b)$ to be divisible by 3.
2. $3a + 3b + 2c = 3(a + b) + 2c$ must be divisible by 3, which requires $2c$ (and hence $c$) to be divisible by 3. Thus, both $(a - b)$ and $c$ must be divisible by 3.