Finding the Unit Digit of a Product

To find the unit digit of a product of numbers, we only need to find the unit digit of each number in the product and then find the unit digit of the result when these individual unit digits are multiplied together. In this question, we need to find the unit digit of $23^{65} \times 36^{94} \times 88^{77}$. This means we need to find the unit digit of $23^{65}$, the unit digit of $36^{94}$, and the unit digit of $88^{77}$, and then multiply these unit digits and find the unit digit of that result.

Calculating the Unit Digit of Numbers Raised to Powers

To find the unit digit of a number raised to a power, we look at the pattern (cycle) of the unit digits when the base number is raised to successive positive integer powers.

Unit Digit of $23^{65}$

The base is 23. The unit digit of the base is 3. Let's look at the pattern of the unit digits of powers of 3:

• $3^1 \rightarrow 3$
• $3^2 \rightarrow 9$
• $3^3 \rightarrow 7$ (since $3^3 = 27$)
• $3^4 \rightarrow 1$ (since $3^4 = 81$)
• $3^5 \rightarrow 3$ (since $3^5 = 243$)

The unit digits repeat in a cycle: 3, 9, 7, 1. The length of this cycle is 4. To find the unit digit of $3^{65}$, we need to find the position in this cycle corresponding to the power 65. We do this by finding the remainder when the exponent (65) is divided by the cycle length (4). The calculation is $65 \div 4$. $65 = 4 \times 16 + 1$. The remainder is 1. This means the unit digit of $3^{65}$ is the same as the 1st unit digit in the cycle, which is 3. So, the unit digit of $23^{65}$ is 3.

Unit Digit of $36^{94}$

The base is 36. The unit digit of the base is 6. Let's look at the pattern of the unit digits of powers of 6:

• $6^1 \rightarrow 6$
• $6^2 \rightarrow 6$ (since $6^2 = 36$)
• $6^3 \rightarrow 6$ (since $6^3 = 216$)

The unit digit of any positive integer power of a number ending in 6 is always 6. So, the unit digit of $36^{94}$ is 6.

Unit Digit of $88^{77}$

The base is 88. The unit digit of the base is 8. Let's look at the pattern of the unit digits of powers of 8:

• $8^1 \rightarrow 8$
• $8^2 \rightarrow 4$ (since $8^2 = 64$)
• $8^3 \rightarrow 2$ (since $8^3 = 512$)
• $8^4 \rightarrow 6$ (since $8^4 = 4096$)
• $8^5 \rightarrow 8$ (since $8^5 = 32768$)

The unit digits repeat in a cycle: 8, 4, 2, 6. The length of this cycle is 4. To find the unit digit of $8^{77}$, we find the remainder when the exponent (77) is divided by the cycle length (4). The calculation is $77 \div 4$. $77 = 4 \times 19 + 1$. The remainder is 1. This means the unit digit of $8^{77}$ is the same as the 1st unit digit in the cycle, which is 8. So, the unit digit of $88^{77}$ is 8.

Finding the Unit Digit of the Final Product

We have the unit digits of the three factors:

• Unit digit of $23^{65}$ is 3.
• Unit digit of $36^{94}$ is 6.
• Unit digit of $88^{77}$ is 8.

Now we find the unit digit of the product of these unit digits: $3 \times 6 \times 8$. First, $3 \times 6 = 18$. The unit digit is 8. Next, we find the unit digit of the product of 8 (from $3 \times 6$) and the remaining unit digit, 8. The unit digit of $18 \times 8$ is the same as the unit digit of $8 \times 8$. $8 \times 8 = 64$. The unit digit of 64 is 4. Therefore, the unit digit of the product $23^{65} \times 36^{94} \times 88^{77}$ is 4.