To determine the unit digit in the sum of $124^{372} + 124^{373}$, we only need to focus on the unit digit of the base $124$, which is $4$. Here's the process:

Step 1: Find the unit digit pattern for $4^n$

The unit digits of powers of $4$ follow a repeating cycle:

• $4^1 = 4$ → Unit digit is $4$
• $4^2 = 16$ → Unit digit is $6$
• $4^3 = 64$ → Unit digit is $4$
• $4^4 = 256$ → Unit digit is $6$

Thus, the unit digit alternates between $4$ and $6$ for even and odd powers of $4$, respectively.

Step 2: Determine the unit digits of $124^{372}$ and $124^{373}$

1. $372$ is even, so $124^{372}$ has the same unit digit as $4^2$, which is $6$.
2. $373$ is odd, so $124^{373}$ has the same unit digit as $4^1$, which is $4$.

Step 3: Add the unit digits

The unit digit of $124^{372} + 124^{373}$ is given by the sum of the unit digits of $6$ and $4$: $6 + 4 = 10$ The unit digit of $10$ is $0$.

Final Answer:

The unit digit of $124^{372} + 124^{373}$ is (d) 0.

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Would you like further explanation or examples? Here are some related questions to expand your understanding:

1. What is the unit digit of $123^{1001} + 123^{1002}$?
2. How do you find the last two digits of a large number like $345^{789}$?
3. What is the unit digit of $7^{25}$?
4. Can the sum of two numbers have the same unit digit as one of the addends? Why or why not?
5. What is the repeating cycle of unit digits for $9^n$?

Tip: Focus on the repeating patterns of unit digits when working with powers to simplify calculations!