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: Understanding the Concept:
\(53 \equiv y \pmod 6\) means that \(53 - y\) is divisible by 6. So \(y\) leaves the same remainder as 53 when divided by 6.

Step 2: Find the remainder.
\(53 = 6 \times 8 + 5\). So the remainder is 5, and \(y \equiv 5 \pmod 6\). Hence \(y = 6k + 5\) for an integer \(k\).

Step 3: Check option 1.
64 divided by 6 leaves remainder 4. So option 1 is wrong.

Step 4: Check option 2.
70 leaves remainder 4, and 81 leaves 3. So option 2 is wrong.

Step 5: Check option 3.
65, 59, 53, 47, 41 all leave remainder 5 and differ by 6 each time. So option 3 is right.

Step 6: Check option 4.
70 leaves remainder 4 and 97 leaves 1. So option 4 is wrong.

Final Answer:
The possible values of y are ...65, 59, 53, 47, 41... \[ \boxed{\text{Option 3}} \]