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.