A value of \( x \) satisfying \( 6x\equiv5\ (\mathrm{mod}\ 7) \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → -5
The congruence \( 6x\equiv5\pmod 7 \) means that 7 divides \( 6x-5 \). For the options,
\( \displaystyle x=-5:\ 6x-5=-35 \)
\( \displaystyle x=3:\ 6x-5=13 \)
\( \displaystyle x=-3:\ 6x-5=-23 \)
\( \displaystyle x=4:\ 6x-5=19 \)
Only \( -35 \) is a multiple of 7, so x = −5 satisfies the congruence.
Shortcut: Since \( 6\equiv-1\pmod 7 \), the congruence becomes \( -x\equiv5 \), so \( x\equiv-5\pmod 7 \) directly.