Find the Remainder

1) The remainder when 3^{31} is divided by 5 is

A) 2 B) 3 C) 4 D) 1

(Hint: - gcd(3,5)=1 and \phi (5)=4, thus 3^4\equiv 1(mod 5).)

2) The reminder when 5^{99} is divided by 13 is

A) 0 B) 8 C) 1 D) 4

 

3) The reminder when 7^{2010} is divided by 25 is

A) 9 B) 18 C) 3 D) 24

 

4) The reminder when 7^{51} is divided by 144 is

A) 49 B) 55 C) 21 D) 143

(Hint: gcd(7,144)=1. \phi (144)=48 )

 

Image Source : Shutterstock

Image Source : Shutterstock

5) What is the value of 123^{241} mod 35?

A) 18 B) 24 C) 32 D) 16

(Hint:  123\equiv 18(mod 35).)

 

6) A number when divided by 765 leaves a remainder 42. What will be the remainder if the number is divided by 17? [Asked in “MAT”]
A) 8 B) 7 C) 6 D) 5

 

7) The remainder, when 15^{23}+23^{23} is divided by 19 is [Asked in “CAT”]
A) 4 B) 15 C) 0 D) 18

 

8) The rightmost non zero digit of the number 30^{2720} is [Asked in “CAT”]
A) 1 B) 3 C) 7 D) 9

 

9) If P=16^3+17^3+18^3+19^3 then P divided by 70 leaves a remainder of [Asked in CAT]
A) 0 B) 1 C) 69 D) 35

[Hint:- P= 35 x (an even number)]

 

10) What are the last two digits of 7^{2008} ? [CAT]
A) 21 B) 61 C) 01 D) 41 E) 81

(Hint:- 7 belongs to U(100) and U(100) is isomorphic to Z_2+Z_{20}, thus 7^{20}=1 in U(100).)

 

11) What are the last three digits of 17^{102} ?

A) 001 B) 017 C) 289 D) 513

(Hint:- 17 belongs to U(1000) and U(1000) is isomorphic to Z_2+Z_2+Z_{100}, thus 17^{100}=1 in U(1000).)

 

12) Let p = 1! + (2 2!) + (3 3!) + ... + (10 10!), then p+2 when divided by 11! leaves a remainder of
A) 10 B) 0 C) 7 D) 1

(Hint:- 1! + (2 2!) = (2-1) 1! + (3-1) 2! = (2! - 1!) + (3! - 2!) = 3! – 1.)
13) The remainder, when 1! + 2! + 3! + ... + 99! + 100! is divided by 12 is
A) 11 B) 9 C) 3 D) 1

 

READ:   Music, Mathematics and Mozart

14) The remainder, when 1^5+2^5+3^5+\dots 99^5+100^5 is divided by 4 is
A) 2 B) 0 C) 1 D) 3

(Hint:- (2n)^5\equiv 0 (mod 4), 1\equiv 1 (mod 4) and 3\equiv -1 (mod 4).)

 

15) The remainder, when 1^4+2^4+3^4+\dots 99^4+100^4 is divided by 4 is
A) 2 B) 0 C) 1 D) 3

 

Answers:

1) A 2) B 3) D 4) B 5) A 6) A 7) C 8) D 9) A 10) C 11) C 12) D 13) B 14) B 15) A

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Pankaj Jyoti Mahanta

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2 Comments
  • maverick
    Posted at 01:31h, 10 February

    Can you publish the solutions to these problems?

    • Gonit Sora
      Posted at 14:44h, 23 February

      The answers have been posted. Please check.