When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it?

Dabae
Jun 7, 2015

#1**+20 **

When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it?

Let the base be X

$$\\A*X^3+B*X^2+A*X+B=555\\\\

AX^3+BX^2+AX+B=555\\\\

X^2(AX+B)+(AX+B)=555\\\\

(X^2+1)(AX+B)=555\\\\

So\;\;X^2+1\;\;$must be a factor of 555$\\\\$$

$${factor}{\left({\mathtt{555}}\right)} = {\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{37}}$$

factors of 555 are 3,5,37,15,111,185,555

5 and 37 would both work so far

5 would be base 2 And base 2 is to little, 37 would give base 6 (X=6) so that is probably correct.

Lets see

$$\\(X^2+1)(AX+B)=555\\\\

(6^2+1)(6A+B)=555\\\\

37(6A+B)=555\\\\

6A+B=15\\\\

If A=1 B=8 \qquad $no good A and B cannot be more than 5$\\\\

If A=2 B=3 \qquad $Great$\\\\

$so our number is $ 2323_6$$

Check

$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}} = {\mathtt{555}}$$

And that is excellent

So $$555_{10}=2323_6$$

Melody
Jun 8, 2015

#1**+20 **

Best Answer

When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it?

Let the base be X

$$\\A*X^3+B*X^2+A*X+B=555\\\\

AX^3+BX^2+AX+B=555\\\\

X^2(AX+B)+(AX+B)=555\\\\

(X^2+1)(AX+B)=555\\\\

So\;\;X^2+1\;\;$must be a factor of 555$\\\\$$

$${factor}{\left({\mathtt{555}}\right)} = {\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{37}}$$

factors of 555 are 3,5,37,15,111,185,555

5 and 37 would both work so far

5 would be base 2 And base 2 is to little, 37 would give base 6 (X=6) so that is probably correct.

Lets see

$$\\(X^2+1)(AX+B)=555\\\\

(6^2+1)(6A+B)=555\\\\

37(6A+B)=555\\\\

6A+B=15\\\\

If A=1 B=8 \qquad $no good A and B cannot be more than 5$\\\\

If A=2 B=3 \qquad $Great$\\\\

$so our number is $ 2323_6$$

Check

$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{3}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{6}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}} = {\mathtt{555}}$$

And that is excellent

So $$555_{10}=2323_6$$

Melody
Jun 8, 2015