# Letters in number bases.

I was given a question in number bases.

3P44(base 5) = 246(base 10) find P

I tried converting 246 to base 5 which I taught will be equivalent to 3P44 an then substitute the 25s place as P, but seems incorrect.

This is kind of beyond my comprehension, searched for similar cases but can't find non, probably I am not using the right term(s).

Please is this expression valid, if so, how do I find the solution in cases like this?

• Where were you "given the question"? Is this the whole question? Is there any missing context, or other questions like this one? – Ethan Bolker Aug 15 '17 at 13:47
• Yeah this is the whole of it, there is another similar one too 764x(base 8) = 1660(base 10) find X. – Eddie Dane Aug 15 '17 at 13:51
• $x +4\times 8+6\times 8^2+7\times 8^3=1660$ gives $x=-2340$ and this makes no sense :) Digits in base 8 are $0,1,2\ldots,7$ – Raffaele Aug 15 '17 at 14:25

## 3 Answers

If P in 3P44 stands for a base 5 digit, then there are only five possibilities for P: 0, 1, 2, 3, 4. Let's say P is 0. Then 3P44 is 399 in decimal, which is clearly greater than 246. Any of the other allowable values for P give numbers larger than 246 still.

For what it's worth, 399 decimal in base 13 is 249. I've also thought about negative and balanced bases.

As stated, the problem has no solution, since$$246=5^3+4\times5^2+4\times5+1$$and therefore $246_{10}=1\,441_5$.

Something is wrong.

$$3P44_{10}=4+4×5+P×5^2+3×5^3$$ $$=25P+399=246$$ this gives a negative value for $P$.

• Moreover, $P$ is not even an integer. – T. Linnell Aug 15 '17 at 13:58