What will be the unit digit in the product of 7^71 × 6^63 × 3^65? (a) 1 (b) 2 (c) 3 (d) 4

视频信息

答案文本 复制

视频字幕 复制

To find the unit digit of a large product, we only need to focus on the unit digits of each factor. Let's examine the pattern of unit digits for powers of 7. We see that 7 to the first power ends in 7, 7 squared ends in 9, 7 cubed ends in 3, and 7 to the fourth power ends in 1. This pattern repeats every 4 powers. Now let's find the unit digit of 7 to the 71st power. We already know the pattern repeats every 4 powers: 7, 9, 3, 1. To find which position 71 corresponds to in this cycle, we divide 71 by 4. This gives us 17 with remainder 3. Since the remainder is 3, we look at the 3rd position in our cycle, which is 3. Therefore, 7 to the 71st power has a unit digit of 3. Next, let's find the unit digits of 6 to the 63rd power and 3 to the 65th power. For powers of 6, there's a simple rule: any power of 6 always ends in 6. So 6 to the 63rd power has unit digit 6. For powers of 3, we need to find the pattern. 3 to the first power ends in 3, 3 squared ends in 9, 3 cubed ends in 7, and 3 to the fourth power ends in 1. This cycle repeats every 4 powers. For 65 divided by 4, we get remainder 1, so 3 to the 65th power has unit digit 3. Now let's calculate the final answer. We found that 7 to the 71st power has unit digit 3, 6 to the 63rd power has unit digit 6, and 3 to the 65th power has unit digit 3. To find the unit digit of the entire product, we multiply these unit digits: 3 times 6 times 3. First, 3 times 6 equals 18, then 18 times 3 equals 54. The unit digit of 54 is 4. Therefore, the answer is option d, which is 4.