To find the units digit of \( x = 633^{24} - 277^{33} + 267^{54} \), we will analyze the units digits of each term separately.
### Step 1: Find the units digit of \( 633^{24} \)
The units digit of \( 633 \) is \( 3 \). We need to find the units digit of \( 3^{24} \).
The pattern of the units digits of powers of \( 3 \) is:
- \( 3^1 = 3 \) (units digit is \( 3 \))
- \( 3^2 = 9 \) (units digit is \( 9 \))
- \( 3^3 = 27 \) (units digit is \( 7 \))
- \( 3^4 = 81 \) (units digit is \( 1 \))
This pattern repeats every 4 terms: \( 3, 9, 7, 1 \).
To determine which units digit corresponds to \( 3^{24} \), we find \( 24 \mod 4 \):
\[
24 \mod 4 = 0
\]
Since the remainder is \( 0 \), the units digit of \( 3^{24} \) corresponds to the units digit of \( 3^4 \), which is \( 1 \).
### Step 2: Find the units digit of \( 277^{33} \)
The units digit of \( 277 \) is \( 7 \). We need to find the units digit of \( 7^{33} \).
The pattern of the units digits of powers of \( 7 \) is:
- \( 7^1 = 7 \) (units digit is \( 7 \))
- \( 7^2 = 49 \) (units digit is \( 9 \))
- \( 7^3 = 343 \) (units digit is \( 3 \))
- \( 7^4 = 2401 \) (units digit is \( 1 \))
This pattern also repeats every 4 terms: \( 7, 9, 3, 1 \).
To determine which units digit corresponds to \( 7^{33} \), we find \( 33 \mod 4 \):
\[
33 \mod 4 = 1
\]
Since the remainder is \( 1 \), the units digit of \( 7^{33} \) corresponds to the units digit of \( 7^1 \), which is \( 7 \).
### Step 3: Find the units digit of \( 267^{54} \)
The units digit of \( 267 \) is \( 7 \). We need to find the units digit of \( 7^{54} \).
Using the same pattern for powers of \( 7 \) as before:
- The pattern is \( 7, 9, 3, 1 \).
To determine which units digit corresponds to \( 7^{54} \), we find \( 54 \mod 4 \):
\[
54 \mod 4 = 2
\]
Since the remainder is \( 2 \), the units digit of \( 7^{54} \) corresponds to the units digit of \( 7^2 \), which is \( 9 \).
### Step 4: Combine the results
Now we can combine the units digits we found:
- Units digit of \( 633^{24} \) is \( 1 \).
- Units digit of \( 277^{33} \) is \( 7 \).
- Units digit of \( 267^{54} \) is \( 9 \).
Now, we calculate:
\[
1 - 7 + 9
\]
Calculating this step by step:
1. \( 1 - 7 = -6 \)
2. Now, adding \( 9 \):
\(-6 + 9 = 3\)
Thus, the units digit of \( x \) is \( 3 \).
### Final Answer
The units digit of \( x \) is \( 3 \).
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