Finding the Unit Digit of a Product

The question asks us to find the unit digit of the product of four numbers: 4, 38, 764, and 1256. To find the unit digit of a large product, we only need to focus on the unit digits of the numbers being multiplied.

Steps to Find the Unit Digit

Here's how we can find the unit digit of the product:

1. Identify the unit digit of each number in the product.
2. Multiply these unit digits together.
3. If the result of the multiplication is a number greater than 9, take its unit digit.
4. Continue multiplying the unit digit obtained from the previous step by the unit digit of the next number in the product.
5. The final unit digit obtained is the unit digit of the entire product.

Applying the Method to the Given Numbers

Let's apply this method to the numbers 4, 38, 764, and 1256.

• The unit digit of 4 is 4.
• The unit digit of 38 is 8.
• The unit digit of 764 is 4.
• The unit digit of 1256 is 6.

Now, we multiply these unit digits step by step:

Step 1: Multiply the unit digits of the first two numbers (4 and 38).

Unit digit of 4 is 4.

Unit digit of 38 is 8.

Product of unit digits: \(4 \times 8 = 32\).

The unit digit of 32 is 2.

Step 2: Multiply the unit digit obtained (2) by the unit digit of the third number (764).

Unit digit of 764 is 4.

Product of unit digits: \(2 \times 4 = 8\).

The unit digit of 8 is 8.

Step 3: Multiply the unit digit obtained (8) by the unit digit of the fourth number (1256).

Unit digit of 1256 is 6.

Product of unit digits: \(8 \times 6 = 48\).

The unit digit of 48 is 8.

Therefore, the unit digit of the product \(4 \times 38 \times 764 \times 1256\) is 8.

Summary of Unit Digit Calculation

The final unit digit is 8.

Revision Table: Unit Digit Concepts

Additional Information on Unit Digits

Understanding unit digits is helpful for quick calculations, especially in competitive exams. The pattern of unit digits when a number is raised to increasing powers is cyclic. For example, the unit digits of powers of 2 are 2, 4, 8, 6, 2, 4, 8, 6, and so on (a cycle of 4). Similarly, powers of other digits also have repeating cycles of unit digits.

For example:

• Powers of 0: Unit digit is always 0.
• Powers of 1: Unit digit is always 1.
• Powers of 5: Unit digit is always 5.
• Powers of 6: Unit digit is always 6.
• Powers of 4: Unit digits alternate between 4 and 6 (4, 6, 4, 6...).
• Powers of 9: Unit digits alternate between 9 and 1 (9, 1, 9, 1...).
• Powers of 2: Unit digits cycle through 2, 4, 8, 6.
• Powers of 3: Unit digits cycle through 3, 9, 7, 1.
• Powers of 7: Unit digits cycle through 7, 9, 3, 1.
• Powers of 8: Unit digits cycle through 8, 4, 2, 6.

This concept of cycles is particularly useful when finding the unit digit of large powers of a number.