Let x = (433) 24 – (377) 38 + (166) 54 . What is the units digit of x?
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Understanding Units Digits in Power Calculations
To find the units digit of an expression like \(x = (433)^{24} - (377)^{38} + (166)^{54}\), we need to determine the units digit of each term separately and then perform the indicated subtraction and addition using only the units digits.
The units digit of a number raised to a power depends only on the units digit of the base and the exponent. This is because the units digit of the result of multiplication is determined solely by the units digits of the numbers being multiplied. The pattern of units digits for powers of a single digit is cyclic.
Calculating the Units Digit of Each Term
Units Digit of \( (433)^{24} \)
The base number is 433, and its units digit is 3. We need to find the pattern of the units digits for powers of 3:
- \(3^1 \to 3\)
- \(3^2 \to 9\)
- \(3^3 \to 27 \to 7\)
- \(3^4 \to 81 \to 1\)
- \(3^5 \to 243 \to 3\)
The pattern of the units digits of powers of 3 is 3, 9, 7, 1. This pattern repeats every 4 powers. The cyclicity is 4.
To find the units digit of \( (433)^{24} \), we look at the exponent, which is 24. We divide the exponent by the cyclicity (4) and find the remainder:
\(24 \div 4\)
The remainder is 0. When the remainder is 0, the units digit is the same as the units digit of the last power in the cycle, which corresponds to the 4th power. The units digit of \(3^4\) is 1.
Therefore, the units digit of \( (433)^{24} \) is 1.
Units Digit of \( (377)^{38} \)
The base number is 377, and its units digit is 7. We look at the pattern of the units digits for powers of 7:
- \(7^1 \to 7\)
- \(7^2 \to 49 \to 9\)
- \(7^3 \to 343 \to 3\)
- \(7^4 \to 2401 \to 1\)
- \(7^5 \to 16807 \to 7\)
The pattern of the units digits of powers of 7 is 7, 9, 3, 1. This pattern repeats every 4 powers. The cyclicity is 4.
To find the units digit of \( (377)^{38} \), we look at the exponent, which is 38. We divide the exponent by the cyclicity (4) and find the remainder:
\(38 \div 4 = 9\) with a remainder of 2.
The remainder is 2. The units digit is the same as the units digit of the 2nd power in the cycle. The units digit of \(7^2\) is 9.
Therefore, the units digit of \( (377)^{38} \) is 9.
Units Digit of \( (166)^{54} \)
The base number is 166, and its units digit is 6. We look at the pattern of the units digits for powers of 6:
- \(6^1 \to 6\)
- \(6^2 \to 36 \to 6\)
The pattern of the units digits of powers of 6 is simply 6. This pattern repeats every 1 power. The cyclicity is 1.
For any power of a number ending in 6, the units digit will always be 6.
Therefore, the units digit of \( (166)^{54} \) is 6.
Combining the Units Digits
Now we combine the units digits we found using the operations in the original expression \(x = (433)^{24} - (377)^{38} + (166)^{54}\).
Units digit of \(x\) = (Units digit of \( (433)^{24} \)) - (Units digit of \( (377)^{38} \)) + (Units digit of \( (166)^{54} \))
Units digit of \(x\) = \(1 - 9 + 6\)
When performing subtraction with units digits, if the first digit is smaller than the second, we consider adding 10 to the first digit, similar to borrowing in standard subtraction.
\(1 - 9\)
This is equivalent to \(11 - 9 = 2\) when considering only the units digit.
Now add the units digit of the third term:
\(2 + 6 = 8\)
The resulting units digit is 8.
Final Units Digit of x
The units digit of \(x = (433)^{24} - (377)^{38} + (166)^{54}\) is 8.
| Term | Units Digit of Base | Exponent | Units Digit Pattern (Cyclicity) | Exponent mod Cyclicity | Units Digit of Term |
|---|---|---|---|---|---|
| \((433)^{24}\) | 3 | 24 | 3, 9, 7, 1 (4) | \(24 \div 4\), Remainder 0 (use 4th power) | 1 |
| \((377)^{38}\) | 7 | 38 | 7, 9, 3, 1 (4) | \(38 \div 4\), Remainder 2 | 9 |
| \((166)^{54}\) | 6 | 54 | 6 (1) | Any exponent (use 1st power) | 6 |
Combining the units digits: \(1 - 9 + 6\)
\((1 - 9) \equiv (11 - 9) \pmod{10} \equiv 2 \pmod{10}\)
\(2 + 6 = 8\)
Revision Table: Units Digit Cyclicity
Understanding the cyclic nature of units digits for powers is crucial for solving such problems.
| Units Digit of Base | Pattern | Cyclicity | Units Digit for Exponent rem 1, 2, 3, 0 (or 4) |
|---|---|---|---|
| 0 | 0 | 1 | 0, 0, 0, 0 |
| 1 | 1 | 1 | 1, 1, 1, 1 |
| 2 | 2, 4, 8, 6 | 4 | 2, 4, 8, 6 |
| 3 | 3, 9, 7, 1 | 4 | 3, 9, 7, 1 |
| 4 | 4, 6 | 2 | 4, 6, 4, 6 |
| 5 | 5 | 1 | 5, 5, 5, 5 |
| 6 | 6 | 1 | 6, 6, 6, 6 |
| 7 | 7, 9, 3, 1 | 4 | 7, 9, 3, 1 |
| 8 | 8, 4, 2, 6 | 4 | 8, 4, 2, 6 |
| 9 | 9, 1 | 2 | 9, 1, 9, 1 |
Additional Information on Units Digit Problems
Units digit problems are common in number theory and quantitative aptitude sections of exams. They test the understanding of number properties and patterns in arithmetic.
- To find the units digit of a product, multiply the units digits of the numbers being multiplied and take the units digit of the result.
- To find the units digit of a sum or difference, add or subtract the units digits and take the units digit of the result, handling borrowing if necessary for subtraction.
- The concept of cyclicity of units digits in powers is a key tool for solving problems involving large exponents.
- For exponents that are multiples of the cyclicity (remainder 0), the units digit is the same as the units digit of the base raised to the power equal to the cyclicity.
Important Questions from Unit Digit
The unit digit in 4 × 38 × 764 × 1256 is:
How many times does the number 3 occur in unit's place for numbers ranging from 1 to 100?
A. 20
B. 11
C. 10
D. 19
Find the unit digit in the given factor (3451) 51 × (531) 43 .
A. 6
B. 4
C. 1
D. 9
What is the unit digit of 178 × 593 + 157?
Find the unit digit of $(123)^{123} \times (347)^{347} \times (568)^{568}$.
