In this post i am going to give the tricks to find the units digit of a given number with exponent.For this you have to know about the cyclicity of a given number.
Let’s take an example to understand this:
Example 1 : find the unit digit of 356.
Solution : Now it’s a big term so we cannot find the last digit by doing 3 x 3 x 3 x 3 x 3……. 56 times so we use the concept of cyclicity
Step 1 : 31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
So now pay attention to the last digits we saw that the last digit repeats itself after a cycle of 4 and the cycle is 3 ,9,7,1,this is called the cyclicity of any number ,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 of last digit ends . And in the next step we will divide the power with the cyclicity.
Why the power is divided by number 4.
We will divide the power with 4 because cycle repeat itself after 4 values, and also we need to find the remainder which tells us the required values to complete the next cycle.
Now the main question was that how much is the last digit of 354
So we know the cycle repeats 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, so in the end the unit digit of 354 is 9.
(Type 2 where power of 2 and 3 digits number is to be considered)
Example 2 : What will be the unit digit of 2445 or 34745
Solution :
Lets take some example to understand it very clearly
We know that unit digit of 3 x 3 = 9
And the unit digit of 453 x 543 = 9
The main purpose of the above expression 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 2445 can be found by 445
So the Cyclicity of 4 is 2 because the cycle of last digit repeats after two values
41 = 4
42 =16
43 = 64
So when we divide 45 with 2 then we will get the remainder as 1 and the last digit will be 4
Now come to the case number second unit digit of 34745
The unit digit of this number can be find by the same method
The Cyclicity of 7 is 4
71 = 7
72 = 49
73 = 343
74 = 2401
So on dividing 45 with 4 , 1 will be the remainder and the last digit would be 7
Type 3 ( where pqr is to be considered )
What will be the last digit of 122345
To find the last digit of this type of number we will start the question from the base the base is 12. It means we will see the cyclicity of 2 because the last digit is depends upon the unit digit of 12. Lets do it step vise step
Before the steps we will write the last digits of
21 = 2
22 = 4
23 = 8
24 = 6
25 = 2
Step 1: Now we know that cyclicity of last digit of 12 i.e 2 is 4 , hence the divide the power of 12 i.e 2345 with 4
Step 2: Now the remainder 2345 /4 will determine the last digit.
Step 3: The remainder will be 3 because we can write remainder of 23 /4 = 3 or -1 and -1 45 / 4 will give us remainder as -1 or 3
Hence in the end the last digit of 122345 is nothing but 123 = 8
Practice question:
1) Find the unit digit of 322545 ?
Answer = 2
2)
754
5238 = unit digit is 4
the explanation for the above is the cyclicity of the 8 is 4 and 754/4 the remainder is 2 .the unit digit of 8^2 is 4.
3)
899
456
738
783
2356 = unit digit is 6.
the cyclicity of 6 is same of the number .so,unit digit anything power of 6 is 6.
i think you all got the clarity about the finding the unit digit of any number with exponents.
Let’s take an example to understand this:
Example 1 : find the unit digit of 356.Solution : Now it’s a big term so we cannot find the last digit by doing 3 x 3 x 3 x 3 x 3……. 56 times so we use the concept of cyclicity
Step 1 : 31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
So now pay attention to the last digits we saw that the last digit repeats itself after a cycle of 4 and the cycle is 3 ,9,7,1,this is called the cyclicity of any number ,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 of last digit ends . And in the next step we will divide the power with the cyclicity.
- if the remainder will be 1 then the unit digit will be 3
- if the remainder will be 2 then the unit digit will be 9
- if the remainder will be 3 then the unit digit will be 7
- if the remainder will be 0 then the unit digit will be 1
Why the power is divided by number 4.
We will divide the power with 4 because cycle repeat itself after 4 values, and also we need to find the remainder which tells us the required values to complete the next cycle.
Now the main question was that how much is the last digit of 354
So we know the cycle repeats 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, so in the end the unit digit of 354 is 9.
(Type 2 where power of 2 and 3 digits number is to be considered)
Example 2 : What will be the unit digit of 2445 or 34745
Solution :
Lets take some example to understand it very clearly
We know that unit digit of 3 x 3 = 9
And the unit digit of 453 x 543 = 9
The main purpose of the above expression 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 2445 can be found by 445
So the Cyclicity of 4 is 2 because the cycle of last digit repeats after two values
41 = 4
42 =16
43 = 64
So when we divide 45 with 2 then we will get the remainder as 1 and the last digit will be 4
Now come to the case number second unit digit of 34745
The unit digit of this number can be find by the same method
The Cyclicity of 7 is 4
71 = 7
72 = 49
73 = 343
74 = 2401
So on dividing 45 with 4 , 1 will be the remainder and the last digit would be 7
Type 3 ( where pqr is to be considered )
What will be the last digit of 122345
To find the last digit of this type of number we will start the question from the base the base is 12. It means we will see the cyclicity of 2 because the last digit is depends upon the unit digit of 12. Lets do it step vise step
Before the steps we will write the last digits of
21 = 2
22 = 4
23 = 8
24 = 6
25 = 2
Step 1: Now we know that cyclicity of last digit of 12 i.e 2 is 4 , hence the divide the power of 12 i.e 2345 with 4
Step 2: Now the remainder 2345 /4 will determine the last digit.
Step 3: The remainder will be 3 because we can write remainder of 23 /4 = 3 or -1 and -1 45 / 4 will give us remainder as -1 or 3
Hence in the end the last digit of 122345 is nothing but 123 = 8
Practice question:
1) Find the unit digit of 322545 ?
Answer = 2
2)
754
5238 = unit digit is 4
the explanation for the above is the cyclicity of the 8 is 4 and 754/4 the remainder is 2 .the unit digit of 8^2 is 4.
3)
899
456
738
783
2356 = unit digit is 6.
the cyclicity of 6 is same of the number .so,unit digit anything power of 6 is 6.
i think you all got the clarity about the finding the unit digit of any number with exponents.
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