If a, b, and c are all positive integers, with 4a b, then which of the following conditions is BOTH NECES

Question

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

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Accepted Answer

(a - b) and c are divisible by 3. The given expression is, ( 3 ) a ( 21 ) ( 3 a − b ) ( 49 ) ( 2 b + c ) 3 \sqrt[3]{(3)^a(21)^{(3a - b)}(49)^{(2b + c)}} 3 ( 3 ) a ( 21 ) ( 3 a − b ) ( 49 ) ( 2 b + c ) ​ and it can also be written as, ( 3 ) a ( 3 ) 3 a − b ( 7 ) 3 a − b ( 7 ) 4 b + 2 c 3 \sqrt[3]{(3)^a\ \left(3\right)^{3a\ -\ b}\ \ \

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

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

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

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