Question:

If 5x3y=134385^x-3^y=13438 and 5x1+3y+1=9686,5^{x-1}+3^{y+1}=9686, then x+yx+y equals?

Updated On: Sep 17, 2024
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The Correct Option is D

Solution and Explanation

Taking the equation 5x1+3y+1=96865x−1+3y+1=9686, the last digit of 5x15x−1 will always be 5 for all positive integral values of x.x.

The power cycle of 3 is:
34k+134k+294k+374k13^{4k+1}≡3^{4k+2}≡9^{4k+3}≡7^{4k}≡1

Clearly, 3y+13y+1 must be in the form of 34k34k as the unit digit of the right-hand side is 6.

We have 34=813^4=81, and 38=65613^8=6561. Also, 968681=96059686−81=9605 and 96866561=3125.9686−6561=3125. Observe that 3125=55.3125=55.

Hence, 5x1=555x−1=55 or x=6x=6 and 3y+1=38y=73y+1=38⇒y=7 (where x=6x=6 and y=7y=7 also satisfies the first equation).

Therefore, x+y=6+7=13.x+y=6+7=13.

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