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If a number is represented as 1234 in base $a$ and as 1020304 in base $b$, what can you say about bases a and b?

Attempt at solution:

If we convert each of the representations of the given number to base 10 and equate the results, we obtain:

$$a^3+2a^2+3a=b^6+2b^4+3b^2$$

I am not certain where to proceed from here.

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    You can say that $a=b^2$. – 2017-02-02
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    Could you be more explicit, as to how you arrived at this conclusion? – 2017-02-02
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    That they are both larger than $4$. – 2017-02-02

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$$a^3+2a^2+3a=b^6+2b^4+3b^2$$

That pretty much closes it. Move to one side, use $x^n-y^n=(x-y)(x^{n-1}+ \cdots)$ then:

$$ (b^2-a)(b^4+b^2a+a^2+2b^2+2a+3) = 0 $$

Note that the second factor is strictly positive, so you are left with $b^2=a$ where $a \gt 4$ (for the digit $4$ to be a valid digit in base $a$).

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    Why does $a$ need to be greater than 4? – 2017-02-02
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    @Analysis15: The answer is very explicit about it - **for the digit $4$ to be a valid digit in base $a$**. – 2017-02-02