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If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8 )(7^10)(9^14)(x^y), then what is the value of x?
(1) z is prime
(2) x is prime
The best way to answer this question is to use the exponential rules to simplify the question stem, then analyze each statement based on the simplified equation.
(3^27)(35^10)(z) = (5^8 )(7^10)(9^14)(x^y) Break up the 35^10 and simplify the 9^14
(3^27)(5^10)(7^10)(z) = (5^8 )(7^10)(3^28 )(x^y) Divide both sides by common terms 5^8, 7^10, 3^27
(5^2)(z) = 3x^y
(1) SUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of 3 to balance the 3 on the right side of the equation. Statement (1) says that z is prime, so z cannot have another factor besides the 3. Therefore z = 3.
I am struggling to understand how (1) can be sufficient:
(5^2)(z) = 3x^y
How can we conclude that z MUST have a factor of 3 to balance the right side of the equation ? Why cant z have another factor besides 3 ?
Thanks in advance for your kind help.
(1) z is prime
(2) x is prime
The best way to answer this question is to use the exponential rules to simplify the question stem, then analyze each statement based on the simplified equation.
(3^27)(35^10)(z) = (5^8 )(7^10)(9^14)(x^y) Break up the 35^10 and simplify the 9^14
(3^27)(5^10)(7^10)(z) = (5^8 )(7^10)(3^28 )(x^y) Divide both sides by common terms 5^8, 7^10, 3^27
(5^2)(z) = 3x^y
(1) SUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of 3 to balance the 3 on the right side of the equation. Statement (1) says that z is prime, so z cannot have another factor besides the 3. Therefore z = 3.
I am struggling to understand how (1) can be sufficient:
(5^2)(z) = 3x^y
How can we conclude that z MUST have a factor of 3 to balance the right side of the equation ? Why cant z have another factor besides 3 ?
Thanks in advance for your kind help.
