Unit Digit Calculation Strategy

To find the unit digit of a product of numbers raised to powers, we only need to consider the unit digits of each number and its power.

• The unit digit of a product is determined by the unit digits of the factors involved.
• We will find the unit digit of each term (\(1^1, 3^3, 5^5, 7^7, 9^9\)) separately.
• Then, we find the unit digit of the product of these unit digits.

Unit Digits of Individual Terms

• Unit digit of \(1^1\): The unit digit of \(1\) raised to any positive integer power is always 1.

  Unit digit = 1.

• Unit digit of \(3^3\): \(3^3 = 3 \times 3 \times 3 = 27\).

  Unit digit = 7.

• Unit digit of \(5^5\): The unit digit of \(5\) raised to any positive integer power is always 5.

  Unit digit = 5.

• Unit digit of \(7^7\): The unit digits of powers of 7 follow a cycle: \(7^1 \rightarrow 7\), \(7^2 \rightarrow 9\), \(7^3 \rightarrow 3\), \(7^4 \rightarrow 1\). The cycle length is 4. To find the unit digit of \(7^7\), we find the remainder of \(7 \div 4\), which is 3. The unit digit corresponds to the 3rd element in the cycle.

  Unit digit = 3.

• Unit digit of \(9^9\): The unit digits of powers of 9 follow a cycle: \(9^1 \rightarrow 9\), \(9^2 \rightarrow 1\). The cycle length is 2. To find the unit digit of \(9^9\), we find the remainder of \(9 \div 2\), which is 1. The unit digit corresponds to the 1st element in the cycle.

  Unit digit = 9.

Unit Digit of the Final Product

Now, we multiply the unit digits found above: 1, 7, 5, 3, 9.

The unit digit of the expression \(1^1 \times 3^3 \times 5^5 \times 7^7 \times 9^9\) is the unit digit of the product \(1 \times 7 \times 5 \times 3 \times 9\).

We can calculate the product of the unit digits:

• \(1 \times 7 = 7\)
• \(7 \times 5 = 35\). The unit digit is 5.
• Since we have encountered a unit digit of 5, and all the remaining numbers to be multiplied (3 and 9) are odd, the final unit digit will be 5. Any integer multiplied by 5 results in a number ending in 0 or 5. If the other factor is odd, the result ends in 5.
• Continuing the calculation for completeness: Unit digit of \((...5) \times 3\) is 5. Unit digit of \((...5) \times 9\) is 5.

Therefore, the unit digit of the number \(1^1 \times 3^3 \times 5^5 \times 7^7 \times 9^9\) is 5.