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# Unit Digit of Numbers in Exponential Form

Manipulating with exponent is very interesting and funny too.

Unit Digit of Numbers in Exponential Form

Manipulating with exponent is very interesting and funny too.

We know that 93 = 9 × 9 × 9 = 729 , thus the unit digit (the last number of expanded form) of 93 is 9. Similarly, 44 is 4 × 4 × 4 × 4 = 256 . Thus the unit digit of 44 is 6.

Can you guess the unit digit of 230116 , 18147 , 554 , 5620 and 929 ?

It is very difficult to find by expanding the exponential form. But, we can try to tell the unit digit by observing some patterns.

Look at the following number pattern.

103 = 10 ×10 ×10 = 1000

104 = 10 ×10 ×10 ×10 = 10000

105 = 10 ×10 ×10 ×10 ×10 = 100000

Thus, multiplying 10 by itself several times, we always get the unit digit as 0. In other words, 10 raised to the power of any number has the unit digit 0. That is, the unit digit of 10x is always 0, for any positive integer x.

This is also true when the base is multiples of 10. Consider,

402 = (4 ×10)2 = 42 ×102

=16 ×100 = 1600

Similarly, 230116 = (23 ×10)116 = 23116 ×10116

Thus, the unit digit of 230116 is 0.

Now, observe that,

15 =1×1×1×1×1=1

16 =1×1×1×1×1×1=1

We learnt to expand 11 as 10+1.

So,( 10 +1)2 =112 =11×11 = 121

Similarly, 131=130 +1= (13×10)+1

[ (13 ×10) +1 ]2 = 1312 = 131 ×131 = 17161

Hence, if the exponential number is in the form 1x or [(multiple of 10) +1]x, then the unit digit is always 1, where x is a positive integer.

Therefore, the unit digit of 18147 is 1.

Similarly, by observing the following patterns, we can conclude that the unit digit of number with base ending with 5 is 5 and number with base ending with 6 is 6.

51=5

52 =5×5=25

53 =25×5=125

61=6

62 =6×6=36

63 =36×6=216

Therefore, the unit digit of 554 = ( 50 + 5)4 is 5 and the unit digit of 5620 = ( 50 + 6)20 is 6.

We conclude that, for the base number whose unit digits are 0,1,5 and 6 the unit digit of a number corresponding to any positive exponent remains unchanged.

Example 3.11

Find the unit digit of the following exponential numbers:

(i) 2523

(ii) 81100

(iii) 4631

Solution

(i) 2523

Unit digit of base 25 is 5 and power is 23

Thus, the unit digit of 2523 is 5.

(ii) 81100

Unit digit of base 81 is 1 and power is 100

Thus, the unit digit of 81100 is 1.

(iii) 4631

Unit digit of base 46 is 6 and power is 31

Thus, the unit digit of 4631 is 6.

Try these

Find the unit digit of the following exponential numbers:

(i) 10621 (ii) 258 (iii) 3118 (iv) 2010

i) 10621

The unit digit of 10621 is 6.

ii) 258

The unit digit of 258 is 5.

iii) 3118

The unit digit of 3118 is 1.

iv) 2010

The unit digit of 2010 is 0.

Look into the following example. Observe the pattern of unit digit when the base is 4.

41=4 (odd power)                                       42=4×4=16 (even power)

43 = 4 × 4 × 4 = 16 × 4 = 64 (odd power)  44 =64×4=256 (even power)

45 = 256×4 =1024 (odd power)                 46 = 1024×4=4096 (even power)

Note that for base ending with 4, the unit digit of the expanded form alternates between 4 and 6. Further we can notice when the power is odd its unit digit is 4 and when the power is even it is 6.

Similarly, when the base unit is 9,

91=9 (odd power)                                              92 =9×9=81 (even power)

93 = 9 × 9 × 9 = 81 × 9 = 729 (odd power)        94 =729×9=6561 (even power)

95 = 6561× 9 = 59049 (odd power)                   96 = 59049 × 9 = 531441 (even power)

Thus, for base ending with 9, the unit digit after the expansion is 9 for odd power and is 1 for even power.

As we have seen in the earlier case, this rule is applicable, when the base is in the form of [(multiple of 10)+4] or [(multiple of 10)+9].

For example consider, 2412

In this, unit digit of base 24 is 4 and power is 12 (even power).

Therefore, unit digit of 2412 is 6.

Similarly consider, 8921

Here unit digit of base 89 is 9 and power is 21 (odd power).

Therefore, unit digit of 8921 is 9.

We conclude that, for base ending with 4, the unit digit of the expanded form is 4 for odd power and is 6 for even power. Similarly, for base ending with 9, the unit digit of the expanded form is 9 for odd power and is 1 for even power. Remember, 4 and 6 are complements of 10. Also, 9 and 1 are complements of 10.

Example 3.12

Find the unit digit of the large numbers:

(i) 47

(ii) 6410

Solution

(i) 47

Unit digit of base 4 is 4 and power is 7 (odd power).

Therefore, unit digit of 47 is 4.

(ii) 6410

Unit digit of base 64 is 4 and power is 10 (even power).

Therefore, unit digit of 6410 is 6.

Try these

Find the unit digit of the following exponential numbers:

(i) 6411

Power 11 (odd power) so, the unit digit is 4.

(ii) 2918

Power 18 (even power) so, the unit digit is 1.

(iii) 7919

Power 19 (odd power) so, the unit digit is 9.

(iv) 10432

Power 32 (even power) so, the unit digit is 6.

Example 3.13

Find the unit digit of the large numbers: (i) 912 (ii) 4917

Solution

(i) 912

Unit digit of base 9 is 9 and power is 12 (even power).

Therefore, unit digit of 912 is 1 .

(ii) 4917

Unit digit of base 49 is 9 and power is 17 (odd power).

Therefore, unit digit of 4917 is 9.

The following activity will help you to find the unit digits of exponent numbers whose base is ending with 2,3,7 and 8.

Activity

Observe the table given below. The numbers in first column, that is 2,3,7 and 8 denotes the unit digit of base of the given exponent number and the numbers in the first row, that is 1,2,3 and 0 stands for the remainder when power is divided by 4. For example, consider 26

Unit digit of base 2 is 2 and power is 6. When the power 6 is divided by 4, we get the remainder as 2.

From the table, we see that 2 and 2 corresponds to 4. Therefore, the unit digit of 26 is 4. To verify, 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64 .

Similarly, consider 11720

Unit digit of base 117 is 7 and power is 20. When the power 20 is divided by 4, we get the remainder as 0.

From the table, we see that 7 and 0 corresponds to 1. Therefore, the unit digit of 11720 is 1.

Now, you can extend this activity for finding the unit digits of any exponent number whose base is ending with any of 2,3,7 or 8.

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