Cyclicity of Numbers - Unit Digits in Powers

Cyclicity of Numbers is the repeating pattern followed by the unit digit of a number when it is raised to successive powers. Since the last digit repeats after a fixed number of powers, identifying this cycle helps quickly determine the unit digit of large exponents without performing the complete calculation.

1. Digits 0, 1, 5, and 6: Here, when each of these digits is raised to any power, the unit digit of the final answer is the number itself.

Examples:

1. 5² = 25: Unit digit is 5, the number itself.
2. 1⁶ = 1: Unit digit is 1, the number itself.
3. 0⁴ = 0: Unit digit is 0, the number itself.
4. 6³ = 216: Unit digit is 6, the number itself.

Question 1: Find the unit digit of 416³⁴⁵.

Answer: Simply find 6³⁴⁵ which will give 6 as a unit digit, hence the unit digit of 416³⁴⁵ is 6.

Question 2: Find the unit digit of 235³⁴⁵⁶⁶.

Answer: Find 5³⁴⁵⁶⁶ which will give 5 as a unit digit, hence the unit digit of 235³⁴⁵⁶⁶ is 5.

2. Digits 4 and 9: Both of these two digits, 4 and 9, have a cyclicity of two different digits as their unit digit.

Examples:

1. 4² = 16: Unit digit is 6.
2. 4³ = 64: Unit digit is 4.
3. 4⁴ = 256: Unit digit is 6.
4. 4⁵ = 1024: Unit digit is 4.
5. 9² = 81: Unit digit is 1.
6. 9³ = 729: Unit digit is 9.

It can be observed that the unit digits 6 and4 are repeating in an odd-even order. So, 4 has a cyclicity of 2. Similar is the case with 9.

It can be generalized as follows:

• 4^odd = 4: If 4 is raised to the power of an odd number, then the unit digit will be 4.
• 4^even = 6: If 4 is raised to the power of an even number, then the unit digit will be 6.
• 9^odd = 9: If 9 is raised to the power of an odd number, then the unit digit will be 9.
• 9^even = 1: If 9 is raised to the power of an even number, then the unit digit will be 1.

Below are some questions based on the above concept:

Question 1: Find the unit digit of 414²³.

Answer: 23 is an odd number, so 4^odd=4, hence the unit digit is 4.

Question 2: Find the unit digit of 29⁸².

Answer: 82 is an even number, so 9^even=1, hence the unit digit is 1.

3. Digits 2, 3, 7, and 8: These numbers have a cyclicity of four different numbers.

Examples:

1. 2¹ = 2: Unit digit is 2.
2. 2² = 4: Unit digit is 4.
3. 2³ = 8: Unit digit is 8.
4. 2⁴ = 16: Unit digit is 6.
5. 2⁵ = 32: Unit digit is 2.
6. 2⁶ = 64: Unit digit is 4.

It can be observed that the unit digits 2, 4, 8, 6 repeats themselves after a period of four numbers. Similarly,

• The cyclicity of 3 has 4 different numbers: 3, 9, 7, 1.
• The cyclicity of 7 has 4 different numbers: 7, 9, 3, 1.
• The cyclicity of 8 has 4 different numbers: 8, 4, 2, 6.

Below are some questions based on the above concept:

Question 1: Find the unit digit of 257³⁴⁵.

Answer: 345 % 4 = 1, so 7¹, hence the unit digit is 7.

Question 2: Find the unit digit of 423⁴³.

Answer: 43 % 4 = 3, so 3³, hence 7 is the unit digit.

Question 3: Find the unit digit of 28¹⁴⁶.

Answer: 146 % 4 = 2, so 8², hence the unit digit is 4.

Cyclicity Table: The concepts discussed above can be summarized as: