Number System - Aptitude MCQ Questions and Solutions with Explanations

61

Let a number of three digits have for its middle digit the sum of the other two digits. Then it is a multiple of :

A) 10 B) 11 C) 19 D) 50

Discuss

Answer & Solution

Answer: Option B

Solution:

We know that a number having x,y,z as its digit, is a multiple of 11,
If z + x - y = 0
Hence, y = z + x

62

If n is a whole number greater than 1, then n² (n² - 1) is always divisible by :

A) 8 B) 10 C) 12 D) 16

Discuss

Answer & Solution

Answer: Option C

Solution:

Let N = n² (n² - 1) = n² (n - 1) (n + 1)
Then, n = 2
⇒ N = 2² × (2 - 1) (2 + 1)
= (4 × 1 × 3) = 12
Hence, the required number is 12

63

The smallest 4-digit number exactly divisible by 7 is :

A) 1001 B) 1007 C) 1101 D) 1108

Discuss

Answer & Solution

Answer: Option A

Solution:

The smallest 4-digit number is 1000
This when divided by 7 leaves 6 as remainder.
∴ 1001 is the smallest 4-digit number exactly divisible by 7

64

How many 3-digit numbers are completely divisible by 6 ?

A) 149 B) 150 C) 151 D) 166

Discuss

Answer & Solution

Answer: Option B

Solution:

3-digit numbers divisible by 6 are 102, 108, 114 ..... , 996
This is an A.P. in which a = 102, d = 6 and T_n = 996
∴ T_n = a + (n - 1) d
⇒ 102 + (n - 1) × 6 = 996
⇒ (n - 1) × 6 = 894
⇒ (n - 1) = 149
⇒ n = 150

65

The difference between two numbers is 1365. When the large number is divided by the smaller one, the quotient is 6 and the remainder is 15. What is the smaller number ?

A) 240 B) 270 C) 295 D) 360

Discuss

Answer & Solution

Answer: Option B

Solution:

Let the smaller number be x
Then, larger number = (x + 1365)
∴ x + 1365 = 6x + 15
⇒ 5x = 1350
⇒ x = 270
Hence, the smaller number = 270

66

(7¹⁹ + 2) is divided by 6. The remainder is :

A) 1 B) 2 C) 3 D) 5

Discuss

Answer & Solution

Answer: Option C

Solution:

$$\eqalign{ & \left( {{7^{19}} + 2} \right) \div 6 \cr & = \frac{{{{\left( {6 + 1} \right)}^{19}} + 2}}{6} \cr & = \frac{{{1^{19}} + 2}}{6} \cr & = 3\,{\text{remainder}} \cr} $$

Alternate
(xⁿ - aⁿ) is divisible by (x - a) for all values of n.
∴ (7¹⁹ - 1¹⁹) is divisible by (7 - 1)
⇒ (7¹⁹ - 1) is divisible by 6
⇒ On dividing (7¹⁹ + 2) by 6, remainder obtained = 3

67

If the square of sum of three positive consecutive natural number exceeds the sum of their squares by 292, then what is the largest of the three number?

A) 5 B) 6 C) 7 D) 8

Discuss

Answer & Solution

Answer: Option D

Solution:

Go through option
[(x + 2) + (x + 1) + x]² = [(x + 2)² + (x + 1)² + x²] + 292
Let x = 3
(5 + 4 + 3)² = 25 + 16 + 9 + 292
144 < 342
Option D, x = 6
(8 + 7 + 6)² = 64 + 49 + 36 + 292
441 = 441
∴ Option D satisfy this equation
So, largest is 8.

68

Two consecutive natural numbers are always . . . . . . . .

A) even numbers B) co-prime numbers C) odd numbers D) prime numbers

Discuss

Answer & Solution

Answer: Option B

Solution:

Two consecutive natural numbers are always co-prime number.

69

The least number that must be subtracted from 1294 so that the remaining number when divided by 9, 11, 13 will leave in each case the same remainder 6 is:

A) 2 B) 3 C) 1 D) 4

Discuss

Answer & Solution

Answer: Option C

Solution:

Remaining number LCM (9, 11, 13) + 6 = 1287 + 6 = 1293
Least number = 1294 - 1293 = 1

70

What least number must be subtracted from 210, so that the sum is completely divisible by 11?

A) 2 B) 3 C) 4 D) 1

Discuss

Answer & Solution

Answer: Option D

Solution:

If 210 is divisible by 11 then remainder becomes 1.
1 is subtracted from 210 so that the sum completely divisible by 11.