Basic Concept of Cyclicity
The concept of cyclicity is used to identify the last digit of the number.Let’s take an example to understand this:
Example 1: Find the unit digit of 354.
Solution: Now it’s a very big term and not easy to calculate but we canfind the last digit by using the concept of cyclicity. We observe powers of 3
31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
So now pay attention to the last digits we can observe that the last digit repeats itself after a cycle of 4 and the cycle is 3, 9, 7, 1 this repetition of numbers after a particular stage is called the cyclicity of numbers. Therefore when we need to find the unit digit of any number like 3n we just need to find the number on which the cycle halts. So we divide power n by 4 to check remainder
- If remainder is 1 then the unit digit will be 3
- If remainder is 2 then the unit digit will be 9
- If remainder is 3 then the unit digit will be 7
- If remainder is 0 then the unit digit will be 1
We divided the power by 4 because cycle repeats itself after 4 values .Now the main question was that how much is the last digit of 354 and we know the cycle repeat itself after 4 so we will divide the 54 with 4,so on dividing 54 by 4 the remainder becomes 2 .Now as we discussed above if the remainder is 2 the last digit would be 9.Hence unit digit of 354 is 9.
Example 2:What will be the unit digit of 34745
Lets observe unit digit of 347 x 347 = 9.
The main purpose of the above is that the unit digit of any multiplication depends upon the unit digit of numbers, whatever is the number big or small the unit digit always depends upon the multiplication of the last digit.
So the last digit of 34745 can be found by calculating last digit of 745
We observe unit digit while calculating powers of 7
71 = 7
72 = 49
73 = 343
74 = 2401
75 = 16807
So on dividing 45 with 4, 1 will be the remainder and the last digit would be 7.