Find-a-b-c-and-d-such-that-abc-1-mod-11-abd-2-mod-11-acd-3-mod-11-bcd-4-mod-11-

Question Number 198519 by cortano12 last updated on 21/Oct/23

Find a, b, c and d such that

{\begin{cases} abc = 1 (\mod 11) \\ abd = 2 (\mod 11) \\ acd = 3 (\mod 11) \\ bcd = 4 (\mod 11) \end{cases}

Answered by HomeAlone last updated on 21/Oct/23

a b + c a + d c + a d = a + b d + c d i s c o m p l i c a t i v e

Answered by Hridiana last updated on 21/Oct/23

a + b = 4

a + c = 4

Answered by AST last updated on 21/Oct/23

a^{2} b^{2} c^{2} d^{3} = {(a b c)}^{2} d^{3} \overset{11}{\equiv} 2 \times 3 \times 4 \overset{11}{\equiv} 2 \Rightarrow d^{3} \overset{11}{\equiv} 2

\Rightarrow d \equiv 7 (m o d 11)

a^{2} {b c d}^{2} = (a b c) a (d^{2}) \overset{11}{\equiv} 2 \times 3 \Rightarrow 49 a \overset{11}{\equiv} 6 \Rightarrow - 6 a \overset{11}{\equiv} 6

\Rightarrow a \equiv 10 (m o d 11)

a b d \overset{11}{\equiv} 2 \Rightarrow 4 b \overset{11}{\equiv} 2 \Rightarrow b \equiv 6 (m o d 11)

a b c \overset{11}{\equiv} 1 \Rightarrow 5 c \overset{11}{\equiv} 1 \Rightarrow c \equiv 9 (m o d 11)

Commented by HomeAlone last updated on 21/Oct/23

good job