Question Number 110595 by Aina Samuel Temidayo last updated on 29/Aug/20
$$\mathrm{Evaluate}\:\mathrm{5}!\bullet\mathrm{6}!\left(\mathrm{mod}\:\mathrm{7}!\right) \\ $$
Commented by Her_Majesty last updated on 29/Aug/20
$${you}'{re}\:{right} \\ $$
Commented by Aina Samuel Temidayo last updated on 29/Aug/20
$$\mathrm{Thanks}\:\mathrm{but}\:\mathrm{this}\:\mathrm{is}\:\mathrm{incorrect}. \\ $$
Commented by Her_Majesty last updated on 29/Aug/20
$$\mathrm{5}!\mathrm{6}!=\left(\mathrm{17}+\mathrm{1}/\mathrm{7}\right)\mathrm{7}!\:\Rightarrow\:{answer}\:{is}\:\mathrm{7}!/\mathrm{7}=\mathrm{6}!=\mathrm{720} \\ $$
Commented by Aina Samuel Temidayo last updated on 29/Aug/20
$$\mathrm{Please}\:\mathrm{I}\:\mathrm{need}\:\mathrm{a}\:\mathrm{well}−\mathrm{detailed}\:\mathrm{solution} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{problem}. \\ $$
Commented by Her_Majesty last updated on 29/Aug/20
$${but}\:{it}\:{is}! \\ $$$$\frac{\mathrm{5}!\mathrm{6}!}{\mathrm{7}!}=\frac{\mathrm{5}!\mathrm{6}!}{\mathrm{6}!×\mathrm{7}}=\frac{\mathrm{5}!}{\mathrm{7}}=\frac{\mathrm{120}}{\mathrm{7}}=\mathrm{17}+\frac{\mathrm{1}}{\mathrm{7}} \\ $$$$\Rightarrow \\ $$$$\mathrm{5}!\mathrm{6}!=\left(\mathrm{17}+\frac{\mathrm{1}}{\mathrm{7}}\right)\mathrm{7}!=\mathrm{17}×\mathrm{7}!+\frac{\mathrm{7}!}{\mathrm{7}}=\mathrm{17}×\mathrm{7}!+\mathrm{6}! \\ $$$$\Rightarrow\:\mathrm{6}!=\mathrm{720}\:{is}\:{the}\:{remainder} \\ $$