Number System - Study Mode

[#226] 1260 ÷ 14 ÷ 9 = ?
Correct Answer

(B) 10

Explanation

Solution: $$eqalign{
& = 1260 div 14 div 9 cr
& = left( {1260 imes frac{1}{{14}} imes frac{1}{9}}
ight) cr
& = 10 cr} $$

[#227] Consider the following statements : 1. If x and y are composite numbers, then x + y is always composite. 2. There does not exist a natural number which is neither prime nor composite. Which of the above statements is/are correct ?
Correct Answer

(D) Neither 1 nor 2

Explanation

Solution: Statement 1. Let x = 4 and y = 15. Then, each one of x and y is a composite number. But, x + y = 19, which is not composite. ∴ Statement 1 is not true. Statement 2. We know that 1 is neither prime nor composite. ∴ Statement 2 is not true. Thus, neither 1 nor 2 is correct.

[#228] There are 2 teams-A and B. If 3 people are shifted from Team A to Team B, then Team B has thrice the number of members than Team A. If 2 people are shifted from Team B to Team A, then Team B has double the number of members than Team A. How many members does Team B have originally?
Correct Answer

(C) 42

Explanation

Solution: $$eqalign{
& frac{{A - 3}}{{B + 3}} = frac{1}{3} cr
& Rightarrow 3A - 9 = B + 3 cr
& Rightarrow 3A - B = 12,......left( 1
ight) cr
& frac{{A + 2}}{{B - 2}} = frac{1}{2} cr
& Rightarrow 2A + 4 = B - 2 cr
& Rightarrow 2A - B = - 6,......left( 2
ight) cr
& { ext{Solving equation}}left( 1
ight){ ext{and}}left( 2
ight) cr
& A = 18 cr
& B = 42 cr
& { ext{Hence, Team B have 42 members}}{ ext{.}} cr} $$

[#229] (91 + 92 + 93 + . . . . . . + 110) is equal to
Correct Answer

(B) 2010

Explanation

Solution: $$eqalign{
& 1 + 2 + 3 + ...... + n = frac{{nleft( {n + 1}
ight)}}{2} cr
& herefore 91 + 92 + ...... + 110 cr
& = left( {1 + 2 + ...... + 110}
ight) - left( {1 + 2 + ...... + 90}
ight) cr
& = frac{{110 imes 111}}{2} - frac{{90 imes 91}}{2} cr
& = 6105 - 4095 cr
& = 2010 cr} $$

[#230] How many numbers are there from 700 to 950 (including both) which are neither divisible by 3 nor by 7?
Correct Answer

(C) 144

Explanation

Solution: Number which is divisible by 3 $$eqalign{
& frac{{700}}{3} - frac{{950}}{3} cr
& = 233 - 316 cr
& = 83......left( { ext{i}}
ight) cr} $$ ⇒ Number which is divisible by 7 $$eqalign{
& frac{{700}}{7} - frac{{950}}{7} cr
& = 100 - 135 cr
& = 35 + 1 cr
& = 36......left( {{ ext{ii}}}
ight) cr} $$ (∴ If 1 st number is totally divisible by 7 then we add +1 in final result) ⇒ Number which is divisible by 21 (LCM of 3 or 7) $$eqalign{
& frac{{700}}{{21}} - frac{{950}}{{21}} cr
& = 33 - 45 cr
& = 12......left( {{ ext{iii}}}
ight) cr} $$ ∴ Number which is divisible by 3 or 7 (i) + (ii) - (iii) (83 + 36) - 12 = 119 - 12 = 107 Total number from 700 to 950 n = (950 - 700) + 1 = 251 ∴ Number which is neither divisible by 3 nor 7 ⇒ 251 - 107 = 144