Number System - Study Mode

[#101] x is five digit number. The digit in ten thousands place is 1. the number formed by its digits in units and ten places is divisible by 4. The sum of all the digits is divisible by 3. If 5 and 7 also divide x, then x will be.
Correct Answer

(D) 10080

Explanation

Solution: Let the digits of x be x = abcde According to the question, x = 1bcde [Given ten thousands place is 1.] Now we can check the options as given that the sum of the all digit is divisible by 3. 10080 is the only number given in the option which satisfies all the given conditions.

[#102] A man sells chocolates which are in the boxes. Only either full box or half a box of chocolates can be purchased from him. A customer comes and buys half the number of boxes which the seller had plus half box more. A second customer comes and purchases half the remaining number of boxes plus half a box. After this the seller is left with no chocolate boxes. How many chocolate boxes the seller had initially?
Correct Answer

(B) 3

Explanation

Solution: The best way to go through the options Let there are initially 3 boxes then, 1 st customer gets = $$frac{3}{2}$$ + $$frac{1}{2}$$ = 2 Remaining boxes = 3 - 2 = 1 2 nd customer = $$frac{1}{2}$$ + $$frac{1}{2}$$ = 1 So, option B is correct.

[#103] If x + y + z = 0, then x 3 + y 3 + z 3 is equal to :
Correct Answer

(B) 3xyz

Explanation

Solution: Given, x + y + z = 0 Cubing both side, (x + y + z) 3 = 0 x 3 + y 3 + z 3 - 3xyz = 0 [using formula] x 3 + y 3 + z 3 = 3xyz

[#104] To write all the page numbers of a book, exactly 136 times digit 1 has been used. Find the number of pages in the book.
Correct Answer

(B) 195

Explanation

Solution: From 1- 99 digit 1 is used 20 times. And From 100 - 199, 1 is used 120 times So, from 1 to 199, 1 is used, 20 + 120 = 140 times We need 136.So leave 199, 198, 197 and 196 Required pages = 195

[#105] Given, N = 98765432109876543210 ..... up to 1000 digits, find the smallest natural number n such that N + n is divisible by 11.
Correct Answer

(D) 5

Explanation

Solution: For a no. to be divisible by 11, Sum(odd digit nos) - Sum(even digit nos) = 0 or divisible by 11 If we look at 9876543210, the difference we get is 5 i.e. [(9 + 7 + 5 + 3 + 1) - (8 + 6 + 4 + 2 + 0) = 5] The series is up to 1000 digit, That means, we get $$frac{{1000}}{{10}}$$xa0 = 100 time 5, then the difference will be 5 × 100 = 500 In order for the difference to be divisible by 11, we need to add 5 and the no will become 505 505 is divisible by 11