What is the digit in the unit’s place of the number represented by 3 98 – 3 89 ?
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Understanding the Unit Digit of \(3^{98} - 3^{89}\)
The question asks for the digit in the unit's place of the number resulting from the subtraction of \(3^{89}\) from \(3^{98}\). To solve this, we need to find the unit digit of \(3^{98}\) and the unit digit of \(3^{89}\) separately, and then determine the unit digit of their difference.
Finding Unit Digits of Powers of 3
The unit digits of the powers of any integer follow a pattern. Let's look at the first few powers of 3:
- \(3^1 = 3\) (Unit digit is 3)
- \(3^2 = 9\) (Unit digit is 9)
- \(3^3 = 27\) (Unit digit is 7)
- \(3^4 = 81\) (Unit digit is 1)
- \(3^5 = 243\) (Unit digit is 3)
- \(3^6 = 729\) (Unit digit is 9)
- \(3^7 = 2187\) (Unit digit is 7)
- \(3^8 = 6561\) (Unit digit is 1)
We can see that the unit digits repeat in a cycle of length 4: 3, 9, 7, 1. To find the unit digit of \(3^n\), we need to determine where in this cycle the \(n\)-th power falls. This can be done by looking at the remainder when the exponent \(n\) is divided by the length of the cycle, which is 4.
The cycle is:
- Exponent \(\pmod 4 = 1 \implies\) Unit digit is 3
- Exponent \(\pmod 4 = 2 \implies\) Unit digit is 9
- Exponent \(\pmod 4 = 3 \implies\) Unit digit is 7
- Exponent \(\pmod 4 = 0 \implies\) Unit digit is 1
Calculating the Unit Digit of \(3^{98}\)
The exponent is 98. We divide 98 by 4 to find the remainder:
\(98 \div 4\)
\(98 = 4 \times 24 + 2\)
The remainder is 2. According to the cycle pattern, if the remainder is 2, the unit digit is 9.
So, the unit digit of \(3^{98}\) is 9.
Calculating the Unit Digit of \(3^{89}\)
The exponent is 89. We divide 89 by 4 to find the remainder:
\(89 \div 4\)
\(89 = 4 \times 22 + 1\)
The remainder is 1. According to the cycle pattern, if the remainder is 1, the unit digit is 3.
So, the unit digit of \(3^{89}\) is 3.
Finding the Unit Digit of the Difference
We need to find the unit digit of \(3^{98} - 3^{89}\). This is equivalent to finding the unit digit of (a number ending in the unit digit of \(3^{98}\)) - (a number ending in the unit digit of \(3^{89}\)).
Unit digit of \(3^{98}\) is 9.
Unit digit of \(3^{89}\) is 3.
We need the unit digit of a number ending in 9 minus a number ending in 3. We can determine the unit digit of the difference by subtracting the unit digits:
\(9 - 3 = 6\)
Therefore, the unit digit of \(3^{98} - 3^{89}\) is 6.
Revision Table: Unit Digits of Powers of 3
| Exponent (n) | \(n \pmod 4\) | Unit Digit of \(3^n\) |
|---|---|---|
| 1 | 1 | 3 |
| 2 | 2 | 9 |
| 3 | 3 | 7 |
| 4 | 0 | 1 |
| 5 | 1 | 3 |
| ... | ... | ... |
Additional Information: Unit Digits of Powers
The concept of cyclic unit digits applies to powers of any integer. The length of the cycle depends on the base number. For example:
- Powers of 2: 2, 4, 8, 6, 2, 4, 8, 6, ... (Cycle length 4)
- Powers of 4: 4, 6, 4, 6, ... (Cycle length 2 - depends on exponent being odd/even)
- Powers of 7: 7, 9, 3, 1, 7, 9, 3, 1, ... (Cycle length 4)
- Powers of 0, 1, 5, 6: The unit digit is always the base itself (Cycle length 1)
Understanding these cycles and using modular arithmetic with the exponent helps determine the unit digit of large powers efficiently.
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