Classical

Numbers & Algebra - Numbers and Algebra MCQ

26:  

What is the greatest number consisting of six digits which on being divided by 6, 7, 8, 9, 10 leaves 4, 5, 6, 7, 8 as remainders respectively?

A.

997920

B.

997918

C.

999999

D.

997922

 
 

Option: B

Explanation :

6 - 4 = 2, 7 - 5 = 2,  8 - 6 = 2,  9 - 7 = 2,  10 -8 = 2
L.C.M. of 6, 7, 8, 9, 10 = 2520;
Greatest number of 6 digits = 999999
2520 x 396 + 2079 = 999999
Remainder = 2079.
Subtract 2079 from 999999, then we get 999999- 2079 = 997920.
Subtract 2 from this number to get required number, which is 997918 and which will give the remainders 4, 5, 6, 7, 8 when divided by 5, 6, 7, 8, 9 respectively.


27:  

 There are N questions in an exam.
For i= 1, 2,..... N ,
there are 2N-1 students who answered i or more questions wrongly. If total number of wrong answers is 8191, N then will be

A.

12

B.

11

C.

10

D.

13

 
 

Option: D

Explanation :

Number of students who answered 1 or more questions wrongly = 2N-1.
Number of students who answered 2 or more questions wrongly = 2N-2
Hence number of students who answered 1 question wrongly = 2N-1- 2N-2 = 2N-2
Similarly it can be shown that number of students who answered 2 questions wrongly = 2N-2 - 2N-3 = 2N-3
Similarly we can find number of students who answered K questions wrongly where K ≥ 3
Hence total number of questions attempted wrongly
s = 2N-2+2(2N-3)+3(2N-4 )+...+(N-1)(20) + N(1) ........(1)
∴ s/2 = 2N-3+ 2(2n-4) +.......+ (N-2)(20) + (N-1)/2 + N/2.......... (2)
Substracting equations (1) from (2)
s/2 = 2N-2 + 2N-3 +...+20+1/2
=> s = 2N + 2N-2+2N-3+...+1 = 2N -1
= 8191
=> N = 13


28:  

In an election for the President, if 261 valid votes are cast, for the 5 contestants then least number of votes a candidate requires to receive to win the election are

A.

53

B.

54

C.

257

D.

72

 
 

Option: A

Explanation :

Worst scenario is when other four get equal number of votes.
Let the winning candidate get x votes.
∴ x > (261-x) / 4
=> x > 52 
x = 53


29:  

If 7x + 6y = 420, x and y are natural numbers, then what can be said about x?

A.

x is always odd

B.

x is always even.

C.

x is even only if y is odd.

D.

x is odd if y is even.

 
 

Option: B

Explanation :

7x + 6y = 420
Equation is of the form:
7x + even number = even number.
∴7x has to be even 
 Hence x has to be even.


30:  

Two new charity organizations C1 and C2 were formed, with x members each, on January 1, 2012. On first day of each, subsequent month, in C1, number of members increases by a certain number a, while in C2, number of members increases in such a way that ratio of the number of members in a month to the preceding month bear a ratio equal to b. On May 1, 2012, both organizations had the same number of members. If a = 20x, then b will be

A.

2

B.

3

C.

2.5

D.

3.5

 
 

Option: B

Explanation :

Number of members in C1 on May 1, 2012 = x + 4ya
Number of members in C2 on May 1, 2012= xb4
x + 4a = xb4 and a = 20x
∴ x(b4 - 81)
As x   ≠ 0, b4 - 81 = 0
∴ b = 3




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