Finding the Unit Digit of an Expression

To find the unit digit of a sum or difference of numbers, we only need to find the unit digit of each number and then perform the operations on those unit digits. We will find the unit digit for each term in the expression \((4)^{31} + (36)^{81} - (243)^{26}\).

Unit Digit of \((4)^{31}\)

Let's look at the pattern of unit digits for powers of 4:

• \(4^1 = 4\) (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)

The pattern of unit digits for powers of 4 is 4, 6, 4, 6, and so on. The unit digit is 4 when the exponent is odd, and the unit digit is 6 when the exponent is even.

In the term \((4)^{31}\), the exponent is 31, which is an odd number. Therefore, the unit digit of \((4)^{31}\) is 4.

Unit Digit of \((36)^{81}\)

For any number ending in 6, any positive integer power of that number will also end in 6. Let's see a few examples:

• \(6^1 = 6\) (Unit digit is 6)
• \(6^2 = 36\) (Unit digit is 6)
• \(6^3 = 216\) (Unit digit is 6)

Since 36 ends in 6, the unit digit of \((36)^{81}\) is 6.

Unit Digit of \((243)^{26}\)

To find the unit digit of \((243)^{26}\), we only need to consider the unit digit of the base, which is 3. Let's look at the pattern of unit digits for 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)

The pattern of unit digits for powers of 3 is 3, 9, 7, 1, and this pattern repeats every 4 powers (the cycle length is 4). To find the unit digit of \((243)^{26}\), we divide the exponent 26 by 4 and look at the remainder.

\(26 \div 4\)

\(26 = 4 \times 6 + 2\)

The remainder is 2. The unit digit of \((243)^{26}\) is the same as the unit digit of \(3^2\), which is 9.

Combining the Unit Digits

Now we perform the operation on the unit digits we found:

• Unit digit of \((4)^{31}\) is 4.
• Unit digit of \((36)^{81}\) is 6.
• Unit digit of \((243)^{26}\) is 9.

We need to find the unit digit of \(4 + 6 - 9\).

\(4 + 6 - 9 = 10 - 9 = 1\).

The unit digit of the entire expression is 1.