Question

If $5^x-3^y=13438$ and $5^{x-1}+3^{y+1}=9686,$ then $x+y$ equals?

• A. 11
• B. 14
• C. 15
• D. 13

Answer 1

Answer: (D)

13

Answer 2

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

The power cycle of 3 is:
$3^{4k+1}≡3^{4k+2}≡9^{4k+3}≡7^{4k}≡1$

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

We have $3^4=81$, and $3^8=6561$. Also, $9686−81=9605$ and $9686−6561=3125.$ Observe that $3125=55.$

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

Therefore, $x+y=6+7=13.$