Mechmeera wrote:If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?
A. 1
B. 2
C. 3
D. 4
E. 6
x = (3�5�7³)/(35^n) = (3�5�
7³)/(5^n *
7^n).
For x to be a positive integer, 7^n must divide into 7³, as indicated by the values in red.
Since n must be a positive integer, we get the following 3 options for 7^n:
7^n = 7¹, with the result that x = (3�5�7³)/(5¹7¹) = 3�5�7².
7^n = 7², with the result that x = (3�5�7³)/(5²7²) = 3�5�7¹.
7^n = 7³, with the result that x = (3�5�7³)/(5³7³) = 3�5³.
The correct answer is
C.
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