PROBLEMS ON FINDING UNIT DIGIT OF THE PRODUCT OF WHOLE NUMBERS

Problem 1 :

What is the unit digit in the product of

(684 x 759 x 413 x 676) ?

Solution :

Product of unit digits of the given whole numbers is 

=  (4 x 9 x 3 x 6)

  =  36 x 18 

Unit digit of the product  =  8.

So, the unit digit of the product of given whole numbers is 8.

Problem 2 :

What is the unit digit in the product

(3547)¹⁵³ x (251)⁷² ?

Solution :

Unit digit of 3547 is 7

Evaluating 7¹⁵³ :

7¹  =  7 (Unit digit is 7)

7²  =  49 (Unit digit is 9)

7³  =  343 (Unit digit is 3)

7⁴  =  2401 (Unit digit is 1)

7⁵  =   2401 x 7 (Unit digit is 7)

Every cycle consists of interval 4. By dividing 153 by 4, we get 1 as remainder. So, the unit digit of 7¹⁵³ is 7.

Unit digit of 251 is 1

Evaluating 1⁷² :

Unit digit of 1⁷² is 1.

So, the unit digit of the given product is 7.

Problem 3 :

What is the unit digit in 264¹⁰² + 264¹⁰³ ?

Solution :

  =  264 ¹⁰² + 264 ¹⁰³ 

  =  264 ¹⁰² (1 + 264)

  =  264 ¹⁰² (265)

Calculating the cyclicity of 4 :

4¹  =  4

4²  =  16

4³  =  64

4⁴  =  256

Every cycle consists of interval 2.By dividing 102 by 2, we will get 0 as remainder. So, the unit digit of 4 ¹⁰² is 6.

6(5)  =  30 (unit digit is 0)

So, the required unit digit is 0.

Problem 4 :

What is the unit digit of 7⁹⁵ - 3⁵⁸ ?

Solution :

Cyclicity of 7 :

7¹  =  7 (Unit digit is 7)

7²  =  49 (Unit digit is 9)

7³  =  343 (Unit digit is 3)

7⁴  =  2401 (Unit digit is 1)

7⁵  =  2401 x 7 (Unit digit is 7)

Every cycle consists of interval 4. 

By dividing 95 by 4, we get 3 as remainder. According to cyclicity of 7, 3 will be the unit digit.

Cyclicity of 3 :

3¹  =  3 (Unit digit is 3)

3²  =  9 (Unit digit is 9)

3³  =  27 (Unit digit is 7)

3⁴  =  81 (Unit digit is 1)

3⁵  =  243 (Unit digit is 3)

Every cycle consists of interval 4. 

By dividing 58 by 4, we get 2 as remainder. According to cyclicity of 3, 9 will be the unit digit.

13 - 9  =  4.

Problem 5 :

What is the unit digit in {6374¹⁷⁹³ x 625³¹⁷ x 341⁴⁹¹} ?

Solution :

Cyclicity of 4 consists of interval 2. By multiplying 5 and 1 itself, we will get the same 5 and 1 as unit digits.

Unit digit of 6374¹⁷⁹³ is 4, the unit digit of 625³¹⁷ is 5 and the unit digit of 341⁴⁹¹ is 1.

Product of unit digits  =  4 x 5 x 1  =  20

Hence the unit digit is 0.

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