In what base is 647 the square of 25?

[RChristenk]

Homework Statement

In what base is $647$ the square of $25$?

Relevant Equations

Knowledge of base conversion

$25 \cdot 25 = 625$ in base $10$, and since $647$ is larger than $625$, the base the question is seeking must be smaller than $10$.

So I tried base $9$ and it turns out $25 \cdot 25 = 647$ in base $9$.

The problem here is I'm just guessing. I'm pretty sure there is a systematic way to write out a equation for this, something along the lines of $6 \cdot r^2 + 4\cdot r + 7 = 25 \cdot 25$ or something. But I don't know how. Thanks for the help.

 

[Hill]

$6 \cdot r^2 + 4\cdot r + 7 = (2 \cdot r+5)^2$

 

[fresh_42]

I get $23^2=6\cdot 9^2+4\cdot 9+7.$

Guessing is not the worst method in this case, because you need the digit seven, which only leaves you with the cases $r\in \{8,9\}.$

 

[jbriggs444]

I get $23^2=6\cdot 9^2+4\cdot 9+7.$

Guessing is not the worst method in this case, because you need the digit seven, which only leaves you with the cases $r\in \{8,9\}.$

I got $r \in \{18,9\}$. Which is, perhaps, what you meant.

Since the last digit is 7, that means that $25_{10}$ is equal to 7 modulo $r$. Which means that $18_{10}$ is equal to 0 modulo $r$. So $r$ must be a factor of $18_{10}$.

But since 7 is a valid digit, $r$ must be at least 8. The only factors of 18 that are greater than or equal to 8 are 9 and 18 itself.

And then, OP had already reasoned that $r$ was less than 10 which only leaves one possibility.

 

[PeroK]

An interesting observation is that in base 7: $25^2 = 1024$, where both numbers are squares base 10.