Question

If \(53 \equiv y \pmod 6\), then the possible values of \(y\) are:-

• A. ......... 64, 58, 52, 46, 40, ........
• B. ......... 70, 76, 81, 87, 93, 98, ........
• C. ......... 65, 59, 53, 47, 41, ........
• D. ......... 70, 76, 82, 88, 94, 97 ........

Hint:

Find \(53 \bmod 6\) and look for numbers with the same remainder.

Answer 1

The Correct Option is C

Step 1: Use the difference form.
The congruence says 6 divides \(53 - y\). So \(53 - y = 6m\), giving \(y = 53 - 6m\).

Step 2: List values.
For \(m = -2, -1, 0, 1, 2, 3\) we get \(y = 65, 59, 53, 47, 41, 35\). The values step by 6 in both directions.

Step 3: Compare with the options.
Only the third list has this pattern: 65, 59, 53, 47, 41. The first list steps by 6 but sits on remainder 4. The other two lists do not step by 6 throughout.

Final Answer:
Option 3 is correct. \[ \boxed{y = 6k + 5} \]