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Consider the equation $(43)_x = (y3)_8$ where $x$ and $y$ are unknown. The number of possible solutions is _____

edited | 3.2k views
+1
plz explain why x>=5 and y<=7 in the given solution?????
+1
Because, when base is 'x', any digit must be less than x.
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thanks

$$(43)_x=(y3)_8$$

Since a number in base$-k$ can only have digits from $0$ to $(k-1)$, we can conclude that: $x \geq 5$ and $y \leq 7$

Now, the original equation, when converted to decimal base gives:

\begin{align}4x^1 + 3x^0 &= y(8^1)+3(8^0)\\[0.8em]4x + 3 &= 8y + 3\\[0.8em]x&=2y\end{align}

So, we have the following constraints:

$$x\geq 5\\y \leq 7\\x=2y\\x,y \text{ are integers}$$

The set of values of $(x,y)$ that satisfy these constraints are:

\begin{align}(x&,y)\\\hline\\(6&,3)\\(8&,4)\\(10&,5)\\(12&,6)\\(14&,7)\end{align}

I am counting 5 pairs of values.

by Veteran (57k points)
selected
+2
Why x>=5, and y<=7???
+9

if  base is r , any digit must be less than r

as in Decimal number, base is 10, digits as 0,1,2,3,4,5,6,7,8,9.

when base is 2,i.e, binary number , digits are {0,1}

+1
Got it
0
For good question

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$x>4$ and $y<8$

(43)=(y3)8

From this equation it is clear x should be greater than 4 and y should be less than 8.

x>=5, and y<=7

Now convert this equation in decimal

=> 4*x+3=y*8+3

4x=8y

ie x=2y

With given constraint (x>=5 && y<=7)

possible values of x and y are (6,3), (8,4),(10,5),(12,6), (14,7)

So possible number of solution is 5.

by Active (1.8k points)

4x+3 = 8y+3

x = 2y  ....Equation 1

but x has to be greater than 4; coz the maximal number it represents is 4

y has to be less than 8; coz the maximal base possible is 8

hence, we get pairs of (x,y) from equation 1 as

(2y, y)

(6, 3)
(8, 4)
(10, 5)
(12, 6)
(14, 7)

by Boss (30.8k points)