# GATE2004-19

3.4k views

If $73_x$ (in base-x number system) is equal to $54_y$ (in base $y$-number system), the possible values of $x$ and $y$ are

1. $8, 16$
2. $10, 12$
3. $9, 13$
4. $8, 11$

edited

$x\times 7 + 3 = 5 \times y + 4 \implies 7x = 5y + 1$.

Only option satisfying this is D.

edited
0

I guess there are infinitely many solutions to this. (Not sure though. I don't know how to prove that there are infinitely many solutions. May be use mathematical induction?)

Some of the solutions are:

 x y 8 11 18 25 43 60 68 95

I just plotted the graph for 5y = 7x - 1 here - link to get these solutions

0

@rohith1001 Yes, there can be infinitely many solutions. The only conditons are that x > 7 and y > 5.

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