## Number System

#### Number System

1. 216 –1 is divisible by

1. ∵ 216 – 1 = (28)2–1
= (28+1) (28–1)

##### Correct Option: C

∵ 216 – 1 = (28)2–1
= (28+1) (28–1)
= (256 + 1) (256 – 1)
216 – 1 = 257 × 255 which is exactly divisible by 17.

1. Which one of the following will completely divide 571 + 572 + 573 ?

1. 571 + 572 + 573 = 571 (1 + 5 + 52)

##### Correct Option: C

571 + 572 + 573 = 571 (1 + 5 + 52)
⇒ 571 + 572 + ⇒ 573 = 571 × which is exactly divisible by 155.
Hence required answer is 155 .

1. The smallest number that must be added to 803642 in order to obtain a multiple of 11 is

1. ∴ The required number = 11 – 4 = 7
2nd Method to solve this question :
Sum of digits at odd places = 2 + 6 + 0 = 8

##### Correct Option: C

∴ The required number = 11 – 4 = 7
2nd Method to solve this question :
Sum of digits at odd places = 2 + 6 + 0 = 8, sum of digits at even places = 4 + 3 + 8 = 15. For divisibility by 11, difference i.e., (15 – 8) = 0 or mutiple of 11.
∴ The required number = 7

1. If [n] denotes the greatest integer < n and (n) denotes the smallest integer > n, where n is any real number, then
 1 1 × 1 1 − 1 1 ÷ 1 1 + (1.5) is 5 5 5 5

1. [n] < n (integer); (n) > n (integer)
∴ Expression

 1 1 × 1 1 − 1 1 ÷ 1 1 + (1.5) = 2 × 1 – 2 ÷ 1 + 2 5 5 5 5

##### Correct Option: B

[n] < n (integer); (n) > n (integer)
∴ Expression

 1 1 × 1 1 − 1 1 ÷ 1 1 + (1.5) = 2 × 1 – 2 ÷ 1 + 2 = 2 5 5 5 5

1. In a division sum, the divisor is 3 times the quotient and 6 times the remainder. If the remainder is 2, then the dividend is

1. According to question ,
Divisor = 6 × Remainder
Divisor = 6 × 2 = 12
Again, Divisor = 3 × quotient

 ∴  Quotient = 12 = 4 3

##### Correct Option: A

According to question ,
Divisor = 6 × Remainder
Divisor = 6 × 2 = 12
Again, Divisor = 3 × quotient

 ∴  Quotient = 12 = 4 3

Dividend = 12 × 4 + 2
Dividend = 48 + 2 = 50