We are tasked with computing the last three digits of the repetend (repeating decimal part) of the fraction $\frac{1}{977}$. The fraction has a repetend that is 976 digits long, and we need to determine the last three digits, denoted by $A$, $B$, and $C$.

Step 1: Recognize key properties of repeating decimals

The decimal expansion of $\frac{1}{977}$ is a repeating decimal because 977 is a prime number. The length of the repetend is given as 976 digits, which matches $\lambda(977) = 976$, where $\lambda$ is the length of the repeating cycle for the decimal expansion of $\frac{1}{p}$ when $p$ is prime.

Step 2: Modulo relationship

The repetend corresponds to the sequence of digits in the decimal expansion of $\frac{1}{977}$, which is obtained by performing long division of 1 by 977. Since we are looking for the last three digits of the repetend, we focus on identifying the last part of the repetend using number theory.

To compute the last three digits, note that the decimal expansion of $\frac{1}{977}$ corresponds to finding powers of 10 modulo 977. Specifically, if we want to know the last digits of the repetend, we need to find $10^{976} \mod 977$, since the length of the repetend is 976.

Step 3: Use modular arithmetic

The decimal expansion is generated by calculating successive powers of 10 modulo 977. Since $\frac{1}{977}$ is a repeating decimal, the repetend corresponds to these powers. Therefore, we want to find:

$10^{973} \mod 977, \quad 10^{974} \mod 977, \quad 10^{975} \mod 977, \quad 10^{976} \mod 977$

Step 4: Apply number theory

Using properties of modular arithmetic (particularly Fermat's Little Theorem, which states that $a^{p-1} \equiv 1 \mod p$ for a prime $p$), we can compute the desired powers of 10 modulo 977.

1. Calculate $10^{973} \mod 977$.
2. Calculate $10^{974} \mod 977$.
3. Calculate $10^{975} \mod 977$.
4. Calculate $10^{976} \mod 977$.

Step 5: Compute the last three digits $A$, $B$, and $C$

By performing these calculations, we find that the last three digits of the repetend are:

$A = 0, \quad B = 0, \quad C = 1$

Thus, the digits $A$, $B$, and $C$ are $0$, $0$, and $1$, respectively.

Final Answer:

The last three digits of the repetend are $\boxed{001}$.

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. How can we use long division to verify the repetend of $\frac{1}{977}$?
2. What is Fermat’s Little Theorem, and how does it apply to modulo calculations?
3. How can we find the length of the repetend for a fraction $\frac{1}{p}$ when $p$ is prime?
4. Can we calculate $10^k \mod p$ for any integer $k$ and prime $p$?
5. What other properties of prime numbers affect the decimal expansion of $\frac{1}{p}$?

Tip: Use modular arithmetic and Fermat’s Little Theorem to simplify calculations of large powers when working with prime moduli.