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Question Number 110895 by Aina Samuel Temidayo last updated on 31/Aug/20 $$\mathrm{How}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{2018}\:\mathrm{be} \\ $$$$\mathrm{expressed}\:\mathrm{as}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two}\:\mathrm{squares}? \\ $$ Answered by ajfour last updated on 31/Aug/20 $$\mathrm{43}^{\mathrm{2}} +\mathrm{13}^{\mathrm{2}}…
Question Number 110783 by Aina Samuel Temidayo last updated on 30/Aug/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{rational}\:\mathrm{numbers} \\ $$$$\mathrm{r},\:\mathrm{0}<\mathrm{r}<\mathrm{1},\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{r}\:\mathrm{is}\:\mathrm{written} \\ $$$$\mathrm{as}\:\mathrm{fraction}\:\mathrm{in}\:\mathrm{lowest}\:\mathrm{term}.\:\mathrm{The} \\ $$$$\mathrm{numerator}\:\mathrm{and}\:\mathrm{demominator}\:\mathrm{have}\:\mathrm{a} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{1000}. \\ $$ Commented by mr…
Question Number 110775 by Aina Samuel Temidayo last updated on 30/Aug/20 $$\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{ab}+\mathrm{1}\mid\mathrm{bc}+\mathrm{1}\:\mathrm{and}\:\mathrm{bc}+\mathrm{1}\mid\mathrm{ca}+\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{ab}+\mathrm{1}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{squares}. \\ $$ Terms of Service Privacy Policy…
Question Number 45221 by malwaan last updated on 10/Oct/18 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{divisors}\:; \\ $$$$\mathrm{if}\:\mathrm{any}\:;\:\mathrm{of}\:\mathrm{16000001} \\ $$ Answered by MJS last updated on 10/Oct/18 $$\mathrm{16000001}=\mathrm{109}×\mathrm{229}×\mathrm{641} \\ $$ Commented…
Question Number 110688 by Aina Samuel Temidayo last updated on 30/Aug/20 $$\mathrm{Let}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{ab}+\mathrm{1}\mid\mathrm{bc}+\mathrm{1}\:\mathrm{and}\:\mathrm{bc}+\mathrm{1}\mid\mathrm{ca}+\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{ab}+\mathrm{1}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{two} \\ $$$$\mathrm{squares}. \\ $$ Terms of Service Privacy Policy…
Question Number 110644 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{The}\:\mathrm{Diophantine}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{1}\:=\mathrm{N}\left(\mathrm{xy}+\mathrm{1}\right)\:\mathrm{has} \\ $$$$\mathrm{infinitely}\:\mathrm{many}\:\mathrm{integer} \\ $$$$\mathrm{solutions}\:\mathrm{if}\:\mathrm{N}\:\mathrm{equals}? \\ $$$$\mathrm{Any}\:\mathrm{help}\:\mathrm{please}? \\ $$ Commented…
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…
Question Number 110591 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{positive} \\ $$$$\mathrm{two}−\mathrm{digit}\:\mathrm{integers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{each}\:\mathrm{of}\:\mathrm{their}\:\mathrm{digits}. \\ $$ Answered by Rasheed.Sindhi last updated on 30/Aug/20…
Question Number 110562 by Aina Samuel Temidayo last updated on 29/Aug/20 $$\mathrm{Let}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{be}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{xy}\neq\mathrm{1},\:\mathrm{x}^{\mathrm{2}} \neq\mathrm{y}\:\mathrm{and}\:\mathrm{y}^{\mathrm{2}} \neq\mathrm{x}. \\ $$$$ \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{p}\mid\mathrm{xy}−\mathrm{1}\:\mathrm{and}\:\mathrm{p}\mid\mathrm{x}^{\mathrm{2}} −\mathrm{y} \\ $$$$\mathrm{then}\:\mathrm{p}\mid\mathrm{y}^{\mathrm{2}} −\mathrm{x}\:\mathrm{where}\:\mathrm{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}. \\…