# Number System: Cyclicity of Numbers

Number System is a method of representing numbers on the number line with the help of a set of symbols and rules. These symbols range from 0-9 and are termed digits. The Number System is used to perform mathematical computations ranging from great scientific calculations to calculations like counting the number of Toys for a Kid or the number of chocolates remaining in the box. Number systems comprise multiple types based on the base value for their digits.

** Cyclicity Of Numbers:** The cyclicity of any number is focused on its unit digit mainly. Every unit digit has its own repetitive pattern when raised to any power. This concept is of tremendous use while solving aptitude problems. The concept of cyclicity of numbers can be learned by figuring out the unit digits of all the single-digit numbers from 0 to 9 when raised to certain powers. These numbers can be broadly classified into three categories listed as follows:

**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. 5 ^ 2 = 25:Unit digit is 5, the number itself.2. 1 ^ 6 = 1:Unit digit is 1, the number itself.3. 0 ^ 4 = 0:Unit digit is 0, the number itself.4. 6 ^ 3 = 216:Unit digit is 6, the number itself.

Below are some questions based on the above concept:

**Question 1: **Find the unit digit of 416^{345}.

**Answer: **Simply find 6^{345} which will give 6 as a unit digit, hence the unit digit of 416^{345} is 6.

**Question 2: **Find the unit digit of 235^{34566}.

**Answer: **Find 5^{34566} which will give 5 as a unit digit, hence the unit digit of 235^{34566} 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. 4 ^ 2 = 16:Unit digit is 6.2. 4 ^ 3 = 64:Unit digit is 4.3. 4 ^ 4 = 256:Unit digit is 6.4. 4 ^ 5 = 1024:Unit digit is 4.5. 9 ^ 2 = 81:Unit digit is 1.6. 9 ^ 3 = 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 414^{23}.

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

**Question 2: **Find the unit digit of 29^{82}.

**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. 2 ^ 1 = 2: Unit digit is 2. 2. 2 ^ 2 = 4: Unit digit is 4. 3. 2 ^ 3 = 8: Unit digit is 8. 4. 2 ^ 4 = 16: Unit digit is 6. 5. 2 ^ 5 = 32: Unit digit is 2. 6. 2 ^ 6 = 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, 4.

Below are some questions based on the above concept:

**Question 1: **Find the unit digit of 257^{345}.

**Answer: **345 % 4 = 1, so 7^{1}, hence the unit digit is 7.

**Question 2:** Find the unit digit of 423^{43}.

**Answer: **43 % 4 = 3, so 3^{3}, hence 7 is the unit digit.

**Question 3: **Find the unit digit of 28^{146}.

**Answer:** 146 % 4 = 2, so 8^{2}, hence the unit digit is 4.

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

Number | Cyclicity | Power Cycle |
---|---|---|

1 | 1 | 1 |

2 | 4 | 2, 4, 8, 6 |

3 | 4 | 3, 9, 7, 1 |

4 | 2 | 4, 6 |

5 | 1 | 5 |

6 | 1 | 6 |

7 | 4 | 7, 9, 3, 1 |

8 | 4 | 8, 4, 2, 6 |

9 | 2 | 9, 1 |

10 | 1 | 0 |