Unit Digit Calculation for Powers

To find the unit digit of large powers, we examine the cyclicity of the unit digits of the base.

Unit Digit of (1373)³⁶

• The unit digit of the base is 3.
• The cycle of unit digits for powers of 3 is: 3, 9, 7, 1. The cycle length is 4.
• We need to find the remainder when the exponent (36) is divided by the cycle length (4).
• Calculation: \(36 \div 4 = 9\) with a remainder of \(0\).
• A remainder of \(0\) corresponds to the last digit in the cycle (the 4^th digit).
• Therefore, the unit digit of (1373)³⁶ is 1.

Unit Digit of (1442)²⁰

• The unit digit of the base is 2.
• The cycle of unit digits for powers of 2 is: 2, 4, 8, 6. The cycle length is 4.
• We need to find the remainder when the exponent (20) is divided by the cycle length (4).
• Calculation: \(20 \div 4 = 5\) with a remainder of \(0\).
• A remainder of \(0\) corresponds to the last digit in the cycle (the 4^th digit).
• Therefore, the unit digit of (1442)²⁰ is 6.

Difference Unit Digit Calculation

We need to find the unit digit of the expression (1373)³⁶ - (1442)²⁰.

• Unit digit of (1373)³⁶ is 1.
• Unit digit of (1442)²⁰ is 6.
• We need the unit digit of (Unit Digit of First Term) - (Unit Digit of Second Term).
• This is the unit digit of \(1 - 6\).
• Since \(1\) is less than \(6\), we borrow from the tens place, making it \(11 - 6\).
• Calculation: \(11 - 6 = 5\).
• The unit digit of the given expression is 5.