# Aptitude - Numbers

Exercise : Numbers - General Questions
26.
The difference between a positive proper fraction and its reciprocal is 9/20. The fraction is:
 3 5
 3 10
 4 5
 4 3
Explanation:

 Let the required fraction be x. Then 1 - x = 9 x 20

 1 - x2 = 9 x 20

20 - 20x2 = 9x

20x2 + 9x - 20 = 0

20x2 + 25x - 16x - 20 = 0

5x(4x + 5) - 4(4x + 5) = 0

(4x + 5)(5x - 4) = 0

 x = 4 5

27.
On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder ?
4
5
6
7
Explanation:

Formula: (Divisor*Quotient) + Remainder = Dividend.

Soln:

(56*Q)+29 = D -------(1)

D%8 = R -------------(2)

From equation(2),

((56*Q)+29)%8 = R.

=> Assume Q = 1.

=> (56+29)%8 = R.

=> 85%8 = R

=> 5 = R.

28.
If n is a natural number, then (6n2 + 6n) is always divisible by:
6 only
6 and 12 both
12 only
by 18 only
Explanation:

(6n2 + 6n) = 6n(n + 1), which is always divisible by 6 and 12 both, since n(n + 1) is always even.

29.
107 x 107 + 93 x 93 = ?
19578
19418
20098
21908
None of these
Explanation:
 107 x 107 + 93 x 93 = (107)2 + (93)2 = (100 + 7)2 + (100 - 7)2 = 2 x [(100)2 + 72]       [Ref: (a + b)2 + (a - b)2 = 2(a2 + b2)] = 20098

30.
What will be remainder when (6767 + 67) is divided by 68 ?
1
63
66
67