To identify the unit digit of a number with some power, we must be aware of cyclicity.
Cyclicity of any number is about the last digit and how they appear in a certain defined manner.
Example 1 :
Let us consider the values of 2^{n}, where n = 1, 2, 3, ...........
2^{1 } = 2
2^{2} = 4
2^{3} = 8
2^{4} = 16
2^{5} = 32
2^{6} = 64
In the above calculations of 2^{n},
We get unit digit 2 in the result of 2^{n}, when n = 1.
Again we get 2 in the unit digit of 2^{n}, when n = 5.
That is, in the fifth term.
So, the cyclicity of 2 is 4.
Example 2 :
Let us consider the values of 3^{n}, where n = 1, 2, 3, ...........
3^{1} = 3
3^{2} = 9
3^{3} = 27
3^{4} = 81
3^{5} = 243
3^{6} = 729
In the above calculations of 3^{n},
We get unit digit 3 in the result of 3^{n}, when n = 1.
Again we get 3 in the unit digit of 3^{n}, when n = 5.
That is, in the fifth term.
So, the cyclicity of 3 is 4.
In the same way, we can get cyclicity of others numbers as shown in the table below.
Number 1 2 3 4 5 6 7 8 9 10 |
Cyclicity of a number 1 4 4 2 1 1 4 4 2 1 |
Example 1 :
Find the unit digit of 32^{24}.
Step 1 :
Take the unit digit in 32 and find its cyclicity.
The unit digit of 32 is '2' and its cyclicity is 4.
Step 2 :
Divide the exponent 24 by the cyclicity 4.
Step 3 :
When 24 is divided by 4, the remainder is zero.
Because the remainder is zero, we can get the unit digit of 2^{8}, from the last value of the cyclicity of 2^{n}_{.}
The last value of the cyclicity of 2^{n} is
2^{4} = 16
The unit digit of 16 is '6'.
Therefore, the unit digit of 32^{24 }is 6.
In step 3 of the above example, what if the remainder is not zero?
It has been explained in the next example.
Example 2 :
Find the unit digit of 32^{27}.
Step 1 :
Take the unit digit in 32 and find its cyclicity.
The unit digit of 32 is '2' and its cyclicity is 4.
Step 2 :
Divide the exponent 27 by the cyclicity 4.
Step 3 :
When 27 is divided by 4, the remainder is 3.
Take the remainder 3 as power of 2 (unit digit of 32).
2^{3} = 8
Therefore, the unit digit of 32^{27 }is 8.
Question 1 :
Find the unit digit of (3547)^{153}_{.}
Solution :
In (3547)^{153}, unit digit is 7.
The cyclicity of 7 is 4. Dividing 153 by 4, we get 1 as remainder.
7^{1} = 7
Therefore, the unit digit of (3547)^{153} is 7.
Question 2 :
Find the unit digit of (264)^{102}_{.}
Solution :
In (264)^{102}, unit digit is 4.
The cyclicity of 4 is 2. Dividing 102 by 2, we get 0 as remainder.
Since the remainder is 0, unit digit will be the last digit of a cyclicity number.
4^{2} = 16 (the unit digit is 6)
Unit digit of 4^{102} is 6.
Therefore, the unit digit of (264)^{102} is 6.
Question 3 :
What is the unit digit in the product
(7)^{105}
Solution :
The unit digit is 7.
The cyclicity of 7 is 4. Dividing 105 by 4, we get 1 as remainder.
So,
7^{1} = 7
The unit digit is 7.
Therefore the unit digit of (7)^{105} is 7.
Question 4 :
Find unit digit of (3^{65} x 6^{59} x 7^{71}).
Solution :
Cyclicity of 3 is 4. Dividing 65 by 4, we get the remainder 1.
Then,
3^{1} = 3
So, the unit digit of 3^{65 }is 3.
Cyclicity of 6 is 1. Dividing 59 by 1, we get the remainder 0.
Then,
6^{1} = 6
So, the unit digit of 6^{59 }is 6.
Cyclicity of 7 is 4. Dividing 71 by 4, we get the remainder 3..
Then,
7^{3} = 343
The unit digit of 343 is '3'.
So, the unit digit of 7^{71 }is also 3.
Product of unit digits :
= 3 x 6 x 3
= 54
The unit digit of the product is 4.
Therefore, the unit digit of (3^{65} x 6^{59} x 7^{71}) is 4.
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