Number System - Study Mode

[#456] The greatest whole number, by which the expression n 4 + 6n 3 + 11n 2 + 6n + 24 is divisible for every natural number n , is -
Correct Answer

(D) 48

Explanation

Solution: $$eqalign{
& { ext{According to the question,}} cr
& {{ ext{n}}^4} + 6{{ ext{n}}^3} + 11{{ ext{n}}^2} + 6{ ext{n + 24}} cr
& { ext{put n = 1}} cr
& = 1 + 6 + 11 + 6 + 24 cr
& = 48 cr
& { ext{put n = 2}} cr
& { ext{ = 16 + 48 + 44 + 12 + 24}} cr
& { ext{ = 144}} cr
& { ext{Clearly it is divisible by 48}} cr} $$

[#457] The sum of all natural numbers from 75 to 97 is -
Correct Answer

(D) 1978

Explanation

Solution: The sum of all natural numbers upto 97 - sum of all natural numbers upto 74 sum of n natural numbers $$eqalign{
& = frac{{nleft( {n + 1}
ight)}}{2},,left( {{ ext{Formula}}}
ight) cr
& = frac{{97 imes 98}}{2} - frac{{74 imes 75}}{2} cr
& = 4753 - 2775 cr
& = 1978 cr} $$

[#458] If $$frac{4}{5}$$ of an estate be worth Rs. 16800, then the value of $$frac{3}{7}$$ of it is -
Correct Answer

(B) Rs. 9000

Explanation

Solution: $$eqalign{
& frac{4}{5}{ ext{ of an estate = 16800}} cr
& { ext{1 of an estate = 16800}} imes frac{5}{4} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 21000 cr
& frac{3}{7}{ ext{ of an estate = 21000}} imes frac{3}{7} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 9000 cr} $$

[#459] Which of the following number is not divided by 18 ?
Correct Answer

(D) 65043

Explanation

Solution: A number will divisible by 18 if it is divisible by 2 and 9 Clearly we can see 65043 is not divisible by 2 Because unit digit of 65043 is 3 so this will not be divisible by 18

[#460] A 9 digit number in which zero does not appear and no digits are repeated has the following properties: The number comprising the left most two digits is divisible by 2, that comprising the left most three digits is divisible by 3, and so on
Correct Answer

(B) 381654729

Explanation

Solution: (a) In 183654729, 1836547 is not divisible by 7 (b) In 381654729, 38 is divisible by 2, 381 is divisible by 3, 3816 is divisible by 4, 38165 is divisible by 5, 381654 is divisible by 6, 3816547 is divisible by 7 38165472 is divisible by 8 and 381654729 is divisible by 9 (c) In 983654721, 983 is not divisible by 3 (d) In 981654723, 9816547 is not divisible by 7