UNIT DIGIT OF NUMBERS IN EXPONENTIAL FORM

Manipulating with exponent is very interesting and funny too.

We know that

9³ = 9 x 9 x 9 = 729,

thus the unit digit (the last number of expanded form) of 9³ is 9.

Similarly,

4⁴ = 4 x 4 x 4 x 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 x 10 x 10 = 1000

10⁴ = 10 x 10 x 10 x 10 = 10000

10⁵ = 10 x 10 x 10 x 10 x 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 10^x is always 0, for any positive integer x. This is also true when the base is multiples of 10.

Consider,

40² = (4 x 10)²

= 4² x 10²

= 16 x 100

= 1600

Similarly,

230¹¹⁶ = (23 x 10)¹¹⁶

= 23¹¹⁶ x 10¹¹⁶

Thus, the unit digit of 230¹¹⁶ is 0.

Now, observe that,

1⁵ = 1 x 1 x 1 x 1 x 1 = 1

1⁶ = 1 x 1 x 1 x 1 x 1 x 1 = 1

We learnt to expand 11 as 10 + 1. So,

(10 + 1)² = 11² = 11 x 11 = 121

Similarly,

131 = 130 + 1

= (13 x 10) + 1

[(13 x 10) + 1]² = 131²

= 131 x 131

= 17161

Hence, if the exponential number is in the form

1^x 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.

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.

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

4¹ = 4 (odd exponent)

4³ = 4 x 4 x 4 = 64 (odd exponent)

4⁵ = 4 x 4 x 4 x 4 x 4 = 1024 (odd exponent)

4² = 4 x 4 = 16 (even exponent)

4⁴ = 4 x 4 x 4 x 4 = 256 (even exponent)

4⁶ = 4 x 4 x 4 x 4 x 4 x 4 = 4096 (even exponent)

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 exponent is odd its unit digit is 4 and when the exponent is even it is 6.

Similarly, when the base unit is 9,

9¹ = 9 (odd exponent)

9³ = 9 x 9 x 9 = 729 (odd exponent)

9⁵ = 9 x 9 x 9 x 9 x 9 = 59049 (odd exponent)

9² = 81 (even exponent)

9⁴ = 9 x 9 x 9 x 9 = 6561 (even exponent)

9⁶ = 9 x 9 x 9 x 9 x 9 x 9 = 531441 (even exponent)

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

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.

Here, unit digit of the base 24 is 4 and the exponent is 12 (even exponent). 

Therefore, unit digit of 24¹² is 6.

Similarly consider, 89²¹.

Here, unit digit of the base 89 is 9 and the exponent is 21 (odd exponent).

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.

Solved Problems

Problems 1-10 : Find the unit digit of the given exponential number.

Problem 1 :

25²³

Solution :

Unit digit of the base 25 is 5 and the exponent is 23.

Therefore, the unit digit of 25²³ is 5.

Problem 2 :

81¹⁰⁰

Solution :

Unit digit of the base 81 is 1 and the exponent is 100.

Therefore, the unit digit of 81¹⁰⁰ is 1.

Problem 3 :

46³¹

Solution :

Unit digit of the base 46 is 6 and the exponent is 31.

Therefore, the unit digit of 46³¹ is 6.

Problem 4 :

106²¹

Solution :

Unit digit of the base 106 is 6 and the exponent is 21.

Therefore, the unit digit of 106²¹ is 6.

Problem 5 :

25⁸

Solution :

Unit digit of the base 25 is 5 and the exponent is 8.

Therefore, the unit digit of 25⁸ is 5.

Problem 6 :

31¹⁸

Solution :

Unit digit of the base 31 is 1 and the exponent is 18.

Therefore, the unit digit of 31¹⁸ is 1.

Problem 7 :

20¹⁰

Solution :

Unit digit of the base 20 is 0 and the exponent is 10.

Therefore, the unit digit of 20¹⁰ is 0.

Problem 8 :

4⁷

Solution :

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

Therefore, the unit digit of 4⁷ is 4.

Problem 9 :

64¹⁰

Solution :

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

Therefore, the unit digit of 64¹⁰ is 6.

Problem 10 :

9¹²

Solution :

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

Therefore, the unit digit of 9¹² is 1.

Problem 11 :

49¹⁷

Solution :

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

Therefore, the unit digit of 49¹⁷ is 9.

Problem 12 :

64¹¹

Solution :

Unit digit of the base 64 is 4 and the exponent is 11 (odd exponent).

Therefore, the unit digit of 64¹¹ is 4.

Problem 13 :

29¹⁸

Solution :

Unit digit of the base 29 is 9 and the exponent is 18 (even exponent).

Therefore, the unit digit of 29¹⁸ is 1.

Problem 14 :

104³²

Solution :

Unit digit of the base 104 is 4 and the exponent is 32 (even exponent).

Therefore, the unit digit of 104³² is 6.

Problem 15 :

79¹⁹

Solution :

Unit digit of the base 79 is 9 and the exponent is 19 (odd exponent).

Therefore, the unit digit of 79¹⁹ is 9.

Activity

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

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 the base 2 is 2 and the exponent is 6. When the exponent 6 is divided by 4, we get the remainder 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 x 2 x 2 x 2 x 2 x 2 = 64

Similarl, consider 117²⁰.

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