Finding the Unit Digit of a Product with Exponents

The question asks for the digit at the unit place in the product $28^{96} \times 26^{92} \times 94^{22}$. To find the unit digit of a product, we only need to find the unit digit of each number being multiplied and then find the unit digit of the product of these unit digits.

In this case, the numbers are $28^{96}$, $26^{92}$, and $94^{22}$. We will find the unit digit of each term separately.

Step 1: Find the Unit Digit of $28^{96}$

The unit digit of $28^{96}$ is determined by the unit digit of the base, which is 8. We look at the pattern (cyclicity) of the unit digits of powers of 8:

• $8^1 = 8$
• $8^2 = 64$ (unit digit is 4)
• $8^3 = 512$ (unit digit is 2)
• $8^4 = 4096$ (unit digit is 6)
• $8^5 = 32768$ (unit digit is 8)

The cycle of unit digits for powers of 8 is 8, 4, 2, 6. The length of this cycle is 4.

To find the unit digit of $8^{96}$, we divide the exponent 96 by the cycle length 4:

\( \frac{96}{4} = 24 \)

Since the remainder is 0 (or 4), the unit digit is the same as the 4th term in the cycle.

Thus, the unit digit of $28^{96}$ is 6.

Step 2: Find the Unit Digit of $26^{92}$

The unit digit of $26^{92}$ is determined by the unit digit of the base, which is 6. Let's look at the pattern of the unit digits of powers of 6:

• $6^1 = 6$
• $6^2 = 36$ (unit digit is 6)
• $6^3 = 216$ (unit digit is 6)

The unit digit of any positive integer power of a number ending in 6 is always 6.

Thus, the unit digit of $26^{92}$ is 6.

Step 3: Find the Unit Digit of $94^{22}$

The unit digit of $94^{22}$ is determined by the unit digit of the base, which is 4. Let's look at the pattern of the unit digits of powers of 4:

• $4^1 = 4$
• $4^2 = 16$ (unit digit is 6)
• $4^3 = 64$ (unit digit is 4)

The cycle of unit digits for powers of 4 is 4, 6. The length of this cycle is 2.

The pattern depends on whether the exponent is odd or even. If the exponent is odd, the unit digit is 4. If the exponent is even, the unit digit is 6.

The exponent here is 22, which is an even number.

Thus, the unit digit of $94^{22}$ is 6.

Step 4: Find the Unit Digit of the Product

The unit digit of the product $28^{96} \times 26^{92} \times 94^{22}$ is the unit digit of the product of the unit digits we found for each term:

Unit digit of $28^{96}$ is 6.

Unit digit of $26^{92}$ is 6.

Unit digit of $94^{22}$ is 6.

We need to find the unit digit of \( 6 \times 6 \times 6 \).

• First, \( 6 \times 6 = 36 \). The unit digit is 6.
• Next, multiply this unit digit by the unit digit of the third term: \( 6 \times 6 = 36 \). The unit digit is 6.

The unit digit of the product is 6.

Let's summarize the unit digits:

The unit digit of the final product is the unit digit of \( 6 \times 6 \times 6 \), which is 6.