Finding the Unit Digit of the Product $3^{999} \times 7^{1000}$

To find the unit digit of the product $3^{999} \times 7^{1000}$, we need to find the unit digit of each factor separately and then multiply them.

Unit Digit of $3^{999}$

The unit digits of powers of 3 follow a cycle: $3^1=3$, $3^2=9$, $3^3=27$ (unit digit 7), $3^4=81$ (unit digit 1), $3^5=243$ (unit digit 3). The cycle is (3, 9, 7, 1) with a length of 4.

To find the unit digit of $3^{999}$, we find the remainder when the exponent 999 is divided by the cycle length 4:

$999 \div 4 = 249 \text{ remainder } 3$

Since the remainder is 3, the unit digit of $3^{999}$ is the 3rd digit in the cycle, which is 7.

Unit Digit of $7^{1000}$

The unit digits of powers of 7 follow a cycle: $7^1=7$, $7^2=49$ (unit digit 9), $7^3=343$ (unit digit 3), $7^4=2401$ (unit digit 1), $7^5=16807$ (unit digit 7). The cycle is (7, 9, 3, 1) with a length of 4.

To find the unit digit of $7^{1000}$, we find the remainder when the exponent 1000 is divided by the cycle length 4:

$1000 \div 4 = 250 \text{ remainder } 0$

When the remainder is 0, the unit digit is the last digit in the cycle, which is 1.

Unit Digit of the Product

The unit digit of the product $3^{999} \times 7^{1000}$ is the unit digit of the product of their unit digits.

Unit digit of $3^{999}$ is 7.

Unit digit of $7^{1000}$ is 1.

The unit digit of the product is the unit digit of $7 \times 1$.

$7 \times 1 = 7$

Therefore, the unit digit of $3^{999} \times 7^{1000}$ is 7.