If the base is odd, the final result will always be odd regardless of the exponent to which it is raised. Similarly, if the base is even, the final result will always be even regardless of the nature of the exponent.
This is based on the fact that exponentiation is repeated multiplication and repeatedly multiplying odd numbers will give you an odd result and vice-versa.
Considering the above, \(3^x\) * \(4^y\) = 177,147 can happen only if \(4^y\) is odd. If \(4^y\) was even, the product can never be 177,147 (an odd number). The only way in which \(4^y\) can be odd is when y=0. Remember, any number (other than ZERO itself) raised to the power of 0 equals 1.
Substituting the value of y=0 in the equation x-y = 11, we get x = 11. It’s not just a coincidence then that \(3^{11}\) = 177,147 ?
An alternative approach to solve this question could be to use the cyclicity of units digits of the power of 3, but only after figuring out that \(4^y\) has to be odd. The cyclicity of unit digits of 3 is 3,9,7 and 1. Since the number we have is a number ending with 7, the power should be of the form 4k+3 and this also points us towards the fact that x=11.
The correct answer option is C.
Hope that helps!
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