To find the unit digit of the expression \(3^{68} \times 4^{73} \times 15^{101}\), we need to determine the unit digit of each factor separately and then multiply them.

Unit Digit of Powers

We analyze the unit digit of each term:

Finding Unit Digit of \(3^{68}\)

• The unit digits of powers of 3 follow a repeating cycle: 3, 9, 7, 1. The cycle length is 4.
• To find the unit digit of \(3^{68}\), we look at the exponent 68. We find the remainder when 68 is divided by the cycle length, 4.
• Calculation: \(68 \div 4 = 17\) with a remainder of 0.
• When the remainder is 0, the unit digit is the last digit in the cycle, which is 1.

Finding Unit Digit of \(4^{73}\)

• The unit digits of powers of 4 follow a repeating cycle: 4, 6. The cycle length is 2.
• To find the unit digit of \(4^{73}\), we look at the exponent 73. We find the remainder when 73 is divided by the cycle length, 2.
• Calculation: \(73 \div 2 = 36\) with a remainder of 1.
• When the remainder is 1, the unit digit is the first digit in the cycle, which is 4.

Finding Unit Digit of \(15^{101}\)

• Any positive integer power of a number ending in 5 will always have a unit digit of 5.
• Therefore, the unit digit of \(15^{101}\) is 5.

Calculate Final Unit Digit

Now, we multiply the unit digits of each factor:

• Unit digit of \(3^{68}\) is 1.
• Unit digit of \(4^{73}\) is 4.
• Unit digit of \(15^{101}\) is 5.

The unit digit of the product \(3^{68} \times 4^{73} \times 15^{101}\) is the unit digit of the product of their unit digits:

Unit digit of \((1 \times 4 \times 5)\)

Calculation: \(1 \times 4 \times 5 = 20\).

The unit digit of 20 is 0.