The set of values of \( x \) satisfying \( x\equiv 12\ (mod\ 5) \) is: (Where \( I \) is set of integer)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \{x:x=12+5\lambda,\ \lambda\in I\} \)
By definition, \( x\equiv a\pmod m \) means that m divides x − a:
\( \displaystyle x\equiv 12\pmod 5\ \Rightarrow\ 5\mid(x-12) \)
\( \displaystyle x-12=5\lambda,\ \lambda\in I \)
\( \displaystyle x=12+5\lambda,\ \lambda\in I \)
Why the other options are incorrect
Option (2) \( \{x:x=5+12\lambda,\ \lambda\in I\} \) — Incorrect. Here x − 5 is a multiple of 12, which is \( x\equiv 5\pmod{12} \); for example x = 5 gives 5 − 12 = −7, not a multiple of 5.
Option (3) \( \{x:x=12\lambda,\ \lambda\in I\} \) — Incorrect. This is \( x\equiv 0\pmod{12} \); x = 0 gives 0 − 12 = −12, not a multiple of 5.
Option (4) \( \{x:x=1+12\lambda,\ \lambda\in I\} \) — Incorrect. This is \( x\equiv 1\pmod{12} \); x = 1 gives 1 − 12 = −11, not a multiple of 5.