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!