Number System

Number System


1. How many prime numbers are there between 2 and 32, and what are they ?
Soln:
There are 10 prime numbers between 2 and 32.
They are 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.


A prime number is a number that can be divided only by 1 and itself.
Ans: 11


2.  Which of the following numbers is exactly divisible by both 9 and 11 ?
(a) 277218 (b) 10098 (c) 12345 (d) 181998


Ans: (b) 10098


Soln,



  • Rule for divisibility by 9: If the sum of digits is divisible by 9, the number is divisible by 9.
    = 1 + 0 + 0 + 9 + 8 = 18 = 18 ÷ 9 = 2 → divisible.

  • Rule for divisibility by 11: If the difference between the sum of digits at even and odd places is 0 or a multiple of 11, the number is divisible by 11.
    = (1 + 0 + 8) − (0 + 9) = 9 − 9 = 0 → divisible.
    Hence, 10098 is divisible by both 9 and 11.


3. Which of the following numbers is NOT divisible by 9 ?
(a) 49104 (b) 77832 (c) 35253 (d) 45390


Ans: (d) 45390


Soln,
Rule for divisibility by 9 → The sum of digits must be divisible by 9.


·   (a) 4 + 9 + 1 + 0 + 4 = 18 → divisible


·   (b) 7 + 7 + 8 + 3 + 2 = 27 → divisible


·   (c) 3 + 5 + 2 + 5 + 3 = 18 → divisible


·   (d) 4 + 5 + 3 + 9 + 0 = 21 → not divisible by 9


Therefore, 45390 is not divisible by 9.


4. Which of the following numbers is NOT divisible by 8 ?
(a) 35792 (b) 35112 (c) 35412 (d) 35552


Ans: (c) 35412


Soln.
Rule for divisibility by 8 → The number formed by the last three digits must be divisible by 8.



  • (a) 792 ÷ 8 = 99 → divisible

  • (b) 112 ÷ 8 = 14 → divisible

  • (c) 412 ÷ 8 = 51.5 → not divisible

  • (d) 552 ÷ 8 = 69 → divisible


Hence, 35412 is not divisible by 8.


5. If a 7-digit number 5045A1B is divisible by 11, then what can be the value of (A + B)? 


(a) 11 (b) 5 (c) 17 (d) 8


Soln.


Rule of 11:


A number is divisible by 11 if
(Sum of digits at odd places – Sum of digits at even places) = 0 or 11 or multiple of 11.


Number: 5 0 4 5 A 1 B


Odd places → 5 + 4 + A + B = 9 + A + B
Even places → 0 + 5 + 1 = 6


Difference = (9 + A + B) – 6 = A + B + 3


For divisibility by 11,
A + B + 3 = 11
⟹ A + B = 8


Ans: (d) 8


6. If number 88p554085k6 is divisible by 72, then what is the value of (3k + 2p)?


(a) 12 (b) 7 (c) 13 (d) 23


Answer: (c) 13


Soln.


If number 88p554085k6 is divisible by 72, then what is the value of (3k + 2p)? (a) 12 (b) 7 (c) 13 (d) 23 simple easy trick


1st→ 72 = 8 × 9


So the number must be divisible by 8 and 9 both.


2nd→ Rule of 8


Look at the last 3 digits → 5k6
A number is divisible by 8 if the last three digits of the number form a number that is divisible by 8.


 536 ÷ 8 = 67 → exact → so k = 3


3rd → Rule of 9


A number is divisible by 9 if the sum of all its digits is divisible by 9.


Sum of all digits = 8 + 8 + p + 5 + 5 + 4 + 0 + 8 + 5 + 3 + 6 = 52 + p


For divisibility by 9 → Next multiple after 52 is 54,


so, 52 + p = 54


  → p = 54 - 52 = 2


4th → Find (3k + 2p)


= 3(3) + 2(2)
= 9 + 4 = 13


Ans: (c) 13


Short Trick:


72 = 8 × 9 → check both rules. Last 3 digits 5k6 → 536 ÷ 8 exact ⇒ k = 3.
Sum of digits = 52 + p → next multiple of 9 = 54 ⇒ p = 2. Now (3k + 2p) = 3×3 + 2×2 = 13.
Ans: (c) 13


7. Which is wrong in the number series below ?  7, 16, 34, 69, 142, 286


Wrong Term: 69


Reason: Series follows ×2 + 2



  • 7 → 7×2+2 = 16

  • 16 → 16×2+2 = 34

  • 34 → 34×2+2 = 70 ⟵ should be 70, not 69

  • 70 → 70×2+2 = 142

  • 142 → 142×2+2 = 286


Corrected series : 7, 16, 34, 70, 142, 286.