If \( 53\equiv y\pmod 6 \), then the possible values of \( y \) are:-
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → ……… 65, 59, 53, 47, 41, ……..
\( 53\equiv y\pmod 6 \) means 6 divides 53 − y, so y leaves the same remainder as 53 on division by 6.
\( 53=6\times 8+5 \)
\( y\equiv 5\pmod 6 \)
So y = 5 + 6k for integers k: …, 41, 47, 53, 59, 65, …
Option (3) lists exactly such values, each differing by 6 and including 53 itself.
Why the other options are incorrect
Option (1) ……… 64, 58, 52, 46, 40, …….. — Incorrect. 64 = 6 × 10 + 4 leaves remainder 4, not 5 (and so does every term of this list).
Option (2) ……… 70, 76, 81, 87, 93, 98, …….. — Incorrect. 70 leaves remainder 4, and the gaps are not all 6 (76 to 81 is 5).
Option (4) ……… 70, 76, 82, 88, 94, 97 …….. — Incorrect. 70 leaves remainder 4, and 94 to 97 is a gap of 3.