Question Number 140008 by ajfour last updated on 03/May/21
$$\:{If}\:{in}\:{a}\:{number}\:{system} \\ $$$$\:\:\mathrm{25}×\mathrm{32}=\mathrm{1163} \\ $$$${find}\:{how}\:{many}\:{digits}\:{are}\:{there} \\ $$$${in}\:{the}\:{number}\:{system}\:{used}. \\ $$
Answered by MJS_new last updated on 03/May/21
$$\left(\mathrm{2}{b}+\mathrm{5}\right)\left(\mathrm{3}{b}+\mathrm{2}\right)={b}^{\mathrm{3}} +{b}^{\mathrm{2}} +\mathrm{6}{b}+\mathrm{3} \\ $$$${b}^{\mathrm{3}} −\mathrm{5}{b}^{\mathrm{2}} −\mathrm{13}{b}−\mathrm{7}=\mathrm{0} \\ $$$$\left({b}−\mathrm{7}\right)\left({b}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\:{b}=\mathrm{7} \\ $$
Commented by ajfour last updated on 03/May/21
$${great}\:{way}\:{sir},\:{thought}\:{of}\:{the} \\ $$$${question}\:{myself},\:{but}\:{your} \\ $$$${method}\:{is}\:{great}. \\ $$
Commented by MJS_new last updated on 03/May/21
$$\mathrm{we}\:\mathrm{only}\:\mathrm{need}\:\mathrm{to}\:\mathrm{try}\:\mathrm{the}\:\mathrm{natural}\:\mathrm{factors}>\mathrm{1}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{constant}\:\mathrm{of}\:\mathrm{the}\:\mathrm{resulting}\:\mathrm{polynome}.\:\mathrm{if}\:\mathrm{there}'\mathrm{s} \\ $$$$\mathrm{no}\:\mathrm{such}\:\mathrm{solution},\:\mathrm{no}\:\mathrm{base}\:\mathrm{exists} \\ $$
Commented by ajfour last updated on 03/May/21
$${i}\:{understand}\:{sir}.. \\ $$