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The problem asks for the least residue of the expression \( -40 - 53 \) modulo 12.

### Step 1: Simplify the expression
\[
-40 - 53 = -93
\]

### Step 2: Find the least residue of \(-93 \mod 12\)

To compute the least residue, we divide \(-93\) by 12 and find the remainder:

\[
\frac{-93}{12} = -7.75
\]

The integer part is \(-8\). Multiply back to get:
\[
-8 \times 12 = -96
\]
Now, subtract this from \(-93\):
\[
-93 - (-96) = 3
\]

Thus, the least residue of \(-93 \mod 12\) is **3**.

The answer is:
\[
\boxed{3}
\]

Would you like any additional details or have further questions on this? Here are some related questions to deepen your understanding:

1. What does the term "least residue" mean in modular arithmetic?
2. How do you handle negative numbers in modular arithmetic?
3. What are the properties of modulus operations in different number systems?
4. How does the modulus operation differ from division with remainders?
5. What are some common applications of modular arithmetic in real-world problems?

**Tip:** When dealing with negative numbers in modular arithmetic, it's helpful to remember that the result is always a non-negative remainder.

Find the least residue of -40 - 53 modulo 12.

Learn how to find the least residue of -40 - 53 modulo 12. This problem covers key concepts in modular arithmetic and how to handle negative numbers. Suitable for students in Grades 10-12.

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