Compute 253 (mod 61) using the square-and-multiply

technique. Show the detail of your work.

We are asked to compute \( 253 \mod 61 \) using the **square-and-multiply** technique, which is typically used for modular exponentiation. However, since this is a direct calculation of a number modulo another (not an exponentiation), we can still apply a similar breakdown for understanding.

The square-and-multiply algorithm is usually used when calculating powers, like \( a^b \mod n \), but in this case, we simplify the problem directly as follows:

### Step-by-Step Calculation

#### Step 1: Express 253 in a manageable form
We want to reduce \( 253 \mod 61 \). First, divide 253 by 61 to get the quotient and remainder.

\[
\frac{253}{61} \approx 4.1475
\]

The quotient is \( 4 \), and we now calculate the remainder:

\[
253 - (61 \times 4) = 253 - 244 = 9
\]

Thus, \( 253 \equiv 9 \mod 61 \).

### Conclusion:
The result of \( 253 \mod 61 \) is \( \boxed{9} \).

Would you like more detailed steps or have any other questions?

Here are 5 related questions for further practice:
1. Compute \( 342 \mod 61 \) using the same method.
2. How would you use the square-and-multiply technique to calculate \( 253^5 \mod 61 \)?
3. What is the result of \( 523 \mod 61 \)?
4. Can you verify the remainder of \( 503 \mod 61 \)?
5. Solve \( 123 \mod 19 \) using any method.

**Tip**: When working with modular arithmetic, reducing numbers early in the process (i.e., simplifying expressions modulo \( n \)) can save you a lot of computation!

Compute 253 (mod 61) using the square-and-multiply technique. Show the detail of your work.

Learn how to compute 253 modulo 61 using detailed steps. This problem covers key concepts in modular arithmetic, including remainder calculation and the division algorithm. Suitable for students in Grades 9-12.

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