Competive Exam

1. Q. What is the remainder when 49 is divided by 9 ? Options: A. 9 B. 0 C. 5 D. 4


Soln: 9×5=45, remainder 49−45 = 4


Ans: D. 4


2. Q: Remainder of 49×65÷9 = ? Options: A. 8 B. 0 C. 5 D. 4


Soln:



  • 49 ÷ 9 , remainder 4

  • 65 ÷ 9 , remainder 2

  • Multiply remainders: 4 × 2 = 8


Ans: Remainder = 8


3. Q. Find reminder (1753×1749×83×171) ÷ 17 Options : A. 8  B. 0 C. 7 D. 17


Soln Simple Trick:


Take remainders when divided by 17 = 1753 is 2, 1749 is 15, 83 is 15, 171 is 1


Now multiply 2×15×15×1 = 450


450 ÷ 17 reminder is 8 


Ans: A. 8


4. Q. “What is the remainder when 1!+2!+3!+4!+5!+⋯+1000 is divided by 8 ?


Soln Simple Trick:



  • Factorials 4! and above are divisible by 8 , remainder 0.

  • Only add 1! + 2! + 3! = 1 + 2 + 6 = 9

  • 9 ÷ 8 = remainder 1


Ans: 1


Note:  Why 9÷8 gives remainder 1



  • Divide 9 by 8:


9÷8=1 quotient, remainder ?



  • 8 * 1 = 8

  • 9 − 8 = 1 , this is the remainder


So, 9 ÷ 8 = 1 remainder 1


Rule:


Dividend = (Divisor × Quotient) + Remainder 


Here:


·         9 ÷ 8 , 9 = 8×1 + 1


This is why the remainders is 1


5. Q. What is the remainder when 1!+2!+3!+4!+5!+⋯+1000 is divided by 12 ?


Soln Simple Trick:



  • Factorials 4! and above are divisible by 12 , remainder 0.

  • Only Add 1! + 2! + 3! = 1 + 2 + 6 = 9

  • 9 ÷ 12 , remainder 9


Note : 


Why 9÷12 gives remainder 9


·         Divide 9 by 12:


9÷12=0 quotient, remainder ?


·         12 × 0 = 0


·         9 − 0 = 9 , this is the remainder


So, 9 ÷ 12 = 0 remainder 9


Rule:


Dividend = (Divisor × Quotient) + Remainder 


Here:


·         9 ÷ 12 , 9 = 12×0 + 9


This is why the remainders is 9.


Ans: 9


6. Q. Find the remainder when 35¹¹³ is divided by 9 ?


Soln Simple Trick:


Find remainder of base ÷ divisor. If remainder = divisor − 1 , treat it as −1.


Check the power:



  • Odd power → use −1

  • Even power → use +1


Convert to positive remainder if needed:



  • If result = −1 , remainder = divisor − 1

  • If result = +1 , remainder = 1


Example: 35¹¹³ ÷ 9



  • 35 ≡ 8 ≡ (9 − 1) ≡ −1 (mod 9)

  • Power = 113 (odd) , use −1

  • −1 mod 9 = 9 − 1 = 8 


Remainder = 8


7. Remainder when 68⁸⁸ ÷ 67 ?


Simple Trick:


68 ÷ 67 → remainder = 1
So, 68¹¹³ ≡ 1¹¹³ ≡ 1 (mod 67)


Remainder = 1


8. Finding the remainder when: