Explain Fi nd t he uni t ’s di gi t i n t he pr oduct
(3^67 × 6^39 × 7^53).
Solution : Clearly, unit digit in 34
 is 1
 Unit digit in 3^64 is 1.
 Unit digit in 3^67 is 7. 
(Unit digit in 1 × 3 × 3 × 3 is 7)
Clearly, unit digit in every power of 6 is 6.
 Unit digit in 6^39 is 6.
Clearly, unit digit in 74
 is 1.
 Unit digit in 7^52 is 1.
 Unit digit in 7^53 is 7. (Unit digit in 1 × 7 is 7)
Unit digit in given product
= Unit digit in (7 × 6 × 7) = 4

To find the unit digit of the product \( 3^{67} \times 6^{39} \times 7^{53} \), we can analyze the unit digits of the individual powers separately, and then multiply them together.

### Step 1: Find the unit digit of \( 3^{67} \)
The unit digit of powers of 3 follows a cyclic pattern:
- \( 3^1 \) has a unit digit of **3**.
- \( 3^2 \) has a unit digit of **9**.
- \( 3^3 \) has a unit digit of **7**.
- \( 3^4 \) has a unit digit of **1**.

This cycle (3, 9, 7, 1) repeats every 4 terms. To determine the unit digit of \( 3^{67} \), divide the exponent 67 by 4 and find the remainder:
\[ 67 \div 4 = 16 \text{ R } 3 \]

Since the remainder is 3, the unit digit of \( 3^{67} \) is the same as the unit digit of \( 3^3 \), which is **7**.

### Step 2: Find the unit digit of \( 6^{39} \)
The unit digit of powers of 6 is always 6, regardless of the exponent:
\[ \text{Unit digit of } 6^{39} \text{ is } 6. \]

### Step 3: Find the unit digit of \( 7^{53} \)
The unit digit of powers of 7 also follows a cyclic pattern:
- \( 7^1 \) has a unit digit of **7**.
- \( 7^2 \) has a unit digit of **9**.
- \( 7^3 \) has a unit digit of **3**.
- \( 7^4 \) has a unit digit of **1**.

This cycle (7, 9, 3, 1) repeats every 4 terms. To determine the unit digit of \( 7^{53} \), divide the exponent 53 by 4 and find the remainder:
\[ 53 \div 4 = 13 \text{ R } 1 \]

Since the remainder is 1, the unit digit of \( 7^{53} \) is the same as the unit digit of \( 7^1 \), which is **7**.

### Step 4: Multiply the unit digits
Now multiply the unit digits obtained from the previous steps:
\[ \text{Unit digit of } (7 \times 6 \times 7) \]
\[ = \text{Unit digit of } 294 \text{ is } 4. \]

### Final Answer
Thus, the unit digit of the product \( 3^{67} \times 6^{39} \times 7^{53} \) is **4**.

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Would you like more details, or do you have any questions about this solution?

Here are five related questions:
1. What is the unit digit of \( 2^{45} \times 5^{23} \times 8^{19} \)?
2. How does the cyclic nature of unit digits affect the calculation of large powers?
3. What is the unit digit of \( 4^{58} \times 9^{29} \times 3^{33} \)?
4. Why do certain numbers have repetitive cycles for their unit digits?
5. Can this method be used for finding the last two digits of a product?

**Tip:** When dealing with large exponents, identifying the cycle of unit digits helps in simplifying the problem significantly.

Learn how to find the unit digit of the product 3^67 * 6^39 * 7^53 using cyclic patterns in unit digits and modular arithmetic. Understand the concept of unit digit cycles and apply it to determine the last digit of large products. Suitable for high school students studying number theory and modular arithmetic.

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