Question

In a certain system of writing of numbers, we write $(12\Phi {)}_4=(124{)}_{\Phi }$, where $\Phi$ is the units digit in the first place and number of digits in that system in the second place. How many values of $\Phi$ are possible?

• A. 1
• B. 3
• C. Infinite

Answer 1

${124}_{\Phi }$ shows that value of $\Phi$ should be 5 or more than 5 (since 4 is a digit used in this system).
$(12\Phi {)}_4$ shows that value of $\Phi$ can be at max 3 (since the system = 4).
Now, this is a contradiction. Therefore, no such value exists.