2 votes 2 votes 1) $9^{1015}mod 21=?$ 2)$11^{999}mod 17=?$ Quantitative Aptitude quantitative-aptitude + – srestha asked Dec 20, 2017 srestha 1.4k views answer comment Share Follow See all 25 Comments See all 25 25 Comments reply Show 22 previous comments Ashwin Kulkarni commented Dec 20, 2017 reply Follow Share One another way : $\frac{9^{1015}}{7*3}$ = $\frac{3^{2030}}{7*3}$ = $\frac{3^{2029}}{7}$ = $\frac{3^{3*676 +1}}{7}$ = $\frac{3^{3*676}*3}{7*3}$ = $\frac{(-1)^{676}*3}{7}$ (because 33 mod 7 = -1) Hence it is $\frac{3}{7}$ = $\frac{9}{21}$ 0 votes 0 votes hs_yadav commented Dec 20, 2017 reply Follow Share a simple method to do this.... euler quotient function:- $\Phi (21)=\Phi(3)*\Phi(7)$ now find :- ((9%21)1015%12 ) %21 ..... 97%21 ....i.e 9 1 votes 1 votes Ashwani Kumar 2 commented Dec 20, 2017 reply Follow Share Pulling out highest common factor from Nr and Dr. is making given expression lighter, it's on you otherwise we can directly apply remainder theorem. Question 2) $\frac{11^{999}}{17}$, here apply directly $\phi(17)= 16$ $\frac{rem(11/17)^{rem(999/16)}}{17}$ $\frac{{-6}^7}{17}= \frac{36*36*36* -6}{17}= \frac{-48}{17}=\frac{-14}{17}=3$ 0 votes 0 votes Please log in or register to add a comment.
2 votes 2 votes 94 mod 21= 9 91015mod21 = (94)253mod21 * 93 mod 21= 9 mod 21 * 93mod21 = 94mod21 = 9 11(17−1)mod17= 1 1116* 62 mod17 * 117 mod17 = 1 * 117 mod17 = 117 mod17 = 112*3 mod 17 * 11mod 17 = 8 * 11 mod 17 = 88 mod 17 = 3 Anu007 answered Dec 20, 2017 Anu007 comment Share Follow See all 6 Comments See all 6 6 Comments reply srestha commented Dec 20, 2017 reply Follow Share 9 is cycle of 2 right? 0 votes 0 votes Anu007 commented Dec 20, 2017 reply Follow Share yes.... 0 votes 0 votes srestha commented Dec 20, 2017 reply Follow Share so why u took cycle of 4? 0 votes 0 votes Anu007 commented Dec 20, 2017 reply Follow Share (94) - (21*312) = 9 0 votes 0 votes srestha commented Dec 20, 2017 reply Follow Share not getting 9 is ans at last we have no need to search 9 rt? 0 votes 0 votes smsubham commented Jan 1, 2018 reply Follow Share @srestha we have to check cycle for 9 mod 21 and not cyclic of 9 alone. 9 mod 21 = 9 9^2 mod 21 = 18 9^3 mod 21 = 15 9^4 mod 21 = 9 So on 0 votes 0 votes Please log in or register to add a comment.