GATE CSE 2008 | Question: 6

Question

Let $r$ denote number system radix. The only value(s) of $r$ that satisfy the equation $\sqrt{121_r}={11}_r$ is/are

• A. decimal $10$
• B. decimal $11$
• C. decimal $10$ and $11$
• D. any value $> 2$

Comments

If there was an option "any value > 1", I would have tanked this question, since I didn't keen on notice that the given equation has 2. Any radix $r$ will have $r$ distinct elements $\{0, 1, \cdots r-1\}$.

Since the given equation has 2, $r \not= 2$

Answer 1

$\sqrt{(121)_{r}}=11_{r}$

$\sqrt{(1\times r^{0})+(2\times r^{^{1}})+(1\times r^{2})}=(1\times r^{0})+(1\times r^{1})$

$\sqrt{(1+r)^{^{2}}}=1+r$

$1+r=1+r$

So any integer $r$ satisfies this but $r$ must be greater than $2$ as we have $2$ in $121$ and radix must be greater than any of the digits. (D) is the most appropriate answer

Comments

Because it is satisfying radix greater than 2 and 11 is subset of greater than 2 therefore 11 is incomplete answer

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is the second line the power of r is not correct by the way answer would not change

 because in the unit place power of r should be 0

but due to symmetry answer will not change

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yes

Answer 2

As we all know radix should be greater than number therefore any value > 2 is right and best answer

D)

Comments

Nice answer.correct logic.

Answer 3

https://drive.google.com/file/d/11XtnKsx8dhKPz18wYS8rMMCSHg7Ez8ar/view?usp=drivesdk

May this help you my way of solving this question 

Option D is appropiate ANY VALUE >2

Comments

i am just not understanding the exact logic behind it beacuse ,, both LHS and RHS are same

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it is so coz 121 is perfect square, so if you keep any base it will satify the equation ,  u can try with different bases and it will always satisfy the equation but since the largest digit here is 2 so at minimum base should be 3 or any base > 2 .