Solved Examples

Example 1: Find the unit digit of 7^123. - Unit digit of base = 7 → cycle {7, 9, 3, 1}, length 4. - 123 ÷ 4 = 30 remainder 3 → take the 3rd term of the cycle = 3 Answer: Unit digit = 3

Example 2: Find the unit digit of 234^567. - Unit digit of base = 4 → cycle {4, 6}, length 2. - 567 ÷ 2 = remainder 1 → take the 1st term = 4 Answer: Unit digit = 4

Example 3: Find the unit digit of 89^89 + 73^73. - 89^89: unit digit of base 9, cycle {9, 1}, length 2. 89 ÷ 2 → remainder 1 → 1st term = 9. - 73^73: unit digit of base 3, cycle {3, 9, 7, 1}, length 4. 73 ÷ 4 → remainder 1 → 1st term = 3. - Sum of unit digits = 9 + 3 = 12 → unit digit of the sum = 2 Answer: Unit digit = 2

Example 4: Find the unit digit of 12^36 × 13^57. - 12^36: unit digit of base 2, cycle {2,4,8,6}, length4. 36÷4 → remainder 0 → take last term = 6. - 13^57: unit digit of base 3, cycle {3,9,7,1}, length4. 57÷4 → remainder1 → take 1st term = 3. - Product of unit digits = 6 × 3 = 18 → unit digit = 8 Answer: Unit digit = 8

Example 5: Find the unit digit of 3^57 + 4^57 + 7^57. - 3^57: cycle {3,9,7,1}, length 4. 57 ÷ 4 = 14 remainder 1 → 1st term = 3. - 4^57: cycle {4,6}, length 2. 57 ÷ 2 = remainder 1 → 1st term = 4. - 7^57: cycle {7,9,3,1}, length 4. 57 ÷ 4 = 14 remainder 1 → 1st term = 7. - Sum of unit digits = 3 + 4 + 7 = 14 → unit digit of the total = 4 Answer: Unit digit = 4

Example 6 (Power-tower edge case): Find the unit digit of 2^(3²). - First simplify the exponent itself: 3² = 9. So the expression is really 2^9. - Unit digit of base 2 → cycle {2,4,8,6}, length 4. 9 ÷ 4 = 2 remainder 1 → 1st term = 2 Answer: Unit digit = 2 (Always fully evaluate a “power on top of a power” exponent first before applying cyclicity — a common trap is to apply the cycle rule to the outer exponent without simplifying it.)