Number System has its application in almost every other topic in mathematics. This very much defines the importance of this topic. Number System mainly includes further sub-topics like HCF and LCM, unit digit, factors, cyclicity, factorials, Euler number, digital root, etc.

To understand the concept of unit digit, we must know the concept of cyclicity . This concept is mainly about the unit digit of a number and its repetitive pattern on being divided by a certain number

The concept of unit digit can be learned by figuring out the unit digits of all the single digit numbers from 0 - 9 when raised to certain powers.

These numbers can be broadly classified into three categories for this purpose:

1. **Digits 0, 1, 5 & 6:** When we observe the behaviour of these digits, they all have the same unit's digit as the number itself when raised to any power, i.e. 0^n = 0, 1^n =1, 5^n = 5, 6^n = 6. Let's apply this concept to the following example.

- 185
^{563}

Answer= 5 - 271
^{6987}

Answer= 1 - 156
^{25369 }

Answer= 6 - 190
^{654789321}

Answer= 0

2. **Digits 4 & 9:** Both these numbers have a cyclicity of only two different digits as their unit's digit.

Let us take a look at how the powers of 4 operate: 4^{1} = __4__,

4^{2} = 1__6__,

4^{3} = 6__4__, and so on.

Hence, the power cycle of 4 contains only 2 numbers 4 & 6, which appear in case of odd and even powers respectively.

4

4

Hence, the power cycle of 4 contains only 2 numbers 4 & 6, which appear in case of odd and even powers respectively.

Likewise, the powers of 9 operate as follows:

9^{1} = __9__,

9^{2} = 8__1__,

9^{3} = 72__9__, and so on.

Hence, the power cycle of 9 also contains only 2 numbers 9 & 1, which appear in case of odd and even powers respectively.

9

9

9

Hence, the power cycle of 9 also contains only 2 numbers 9 & 1, which appear in case of odd and even powers respectively.

So, broadly these can be remembered in even and odd only, i.e. 4^{odd} = 4 and 4^{even} = 6. Likewise, 9^{odd} = 9 and 9^{even} = 1.

- 189
^{562589743 }

Answer = 9 (since power is odd) - 279
^{698745832 }

Answer = 1(since power is even) - 154
^{258741369 }

Answer = 4 (since power is odd) - 194
^{65478932}

Answer = 6 (since power is even)

3. **Digits 2, 3, 7 & 8: **These numbers have a power cycle of 4 different numbers.

2^{1} = __2__, 2^{2} = __4__, 2^{3} = __8__ & 2^{4} = 1__6__ and after that it starts repeating.

So, the cyclicity of 2 has 4 different numbers 2, 4, 8, 6.

So, the cyclicity of 2 has 4 different numbers 2, 4, 8, 6.

3^{1} = __3__, 3^{2} =__ 9__, 3^{3} = 2__7__ & 3^{4} = 8__1__ and after that it starts repeating.

So, the cyclicity of 3 has 4 different numbers 3, 9, 7, 1.

So, the cyclicity of 3 has 4 different numbers 3, 9, 7, 1.

7 and 8 follow similar logic.

So these four digits i.e. 2, 3, 7 and 8 have a unit digit cyclicity of four steps.

Must Read Number System Articles

- Number System: Concept of Unit Digit
- Number System: Cyclicity of Remainders

The concepts discussed above are summarized in the given table.

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 |

Suggested Action:

Step 2: Divide the power 562581 by 4.

By doing that, we get a remainder=1.

Step 3: 1

Hence, the answer is 7.

- You must remember the power cycles of all the digits from 1-10.
- If the power cycle of number has 4 different digits, divide the power by 4, find the remaining power and calculate the unit’s digit using that. Similarly, if the power cycle of number has 2 different digits, divide the power by 2, find the remaining power and calculate the unit’s digit using that.