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 7^{153 }:
7^{1 } = 7 (Unit digit is 7)
7^{2} = 49 (Unit digit is 9)
7^{3} = 343 (Unit digit is 3)
7^{4 } = 2401 (Unit digit is 1)
7^{5 } = 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 7^{153} is 7.
Unit digit of 251 is 1
Evaluating 1^{72 }:
Unit digit of 1^{72 }is 1.
So, the unit digit of the given product is 7.
Problem 3 :
What is the unit digit in 264^{102} + 264^{103} ?
Solution :
= 264 ^{102} + 264 ^{103}
= 264 ^{102} (1 + 264)
= 264^{ 102} (265)
Calculating the cyclicity of 4 :
4^{1} = 4
4^{2} = 16
4^{3} = 64
4^{4} = 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 7^{95} - 3^{58} ?
Solution :
Cyclicity of 7 :
7^{1 }= 7 (Unit digit is 7)
7^{2} = 49 (Unit digit is 9)
7^{3} = 343 (Unit digit is 3)
7^{4 }= 2401 (Unit digit is 1)
7^{5 }= 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 :
3^{1 }= 3 (Unit digit is 3)
3^{2} = 9 (Unit digit is 9)
3^{3} = 27 (Unit digit is 7)
3^{4 }= 81 (Unit digit is 1)
3^{5 }= 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 {6374^{1793} x 625^{317} x 341^{491}} ?
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 6374^{1793 }is 4, the unit digit of 625^{317 }is 5 and the unit digit of 341^{491 }is 1.
Product of unit digits = 4 x 5 x 1 = 20
Hence the unit digit is 0.
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