Number System - Aptitude MCQ Questions and Solutions with Explanations

31

The number of times 99 is subtracted from 1111 so that the remainder is less then 99 is :

A) 10 B) 11 C) 12 D) 13

Discuss

Answer & Solution

Answer: Option B

Solution:

On dividing 1111 by 99, the quotient is 11 and the remainder is 22.
Hence, the required number is 11
Number System mcq solution image

32

If p is a positive fraction less than 1, then

A) $$\frac{1}{p}$$ is less than 1 B) $$\frac{1}{p}$$ is a positive integers C) p² is less than p D) $$\frac{2}{p}$$ - p is a positive number

Discuss

Answer & Solution

Answer: Option D

Solution:

p < 1
⇒ $$\frac{1}{p}$$ > 1
⇒ $$\frac{2}{p}$$ > 2
⇒ $$\frac{2}{p}$$ - p > 2 - p > 0 [∵ p < 1]
Hence, $$\left( {\frac{2}{p} - p} \right)$$ is a positive number

33

Given that (1² + 2² + 3² + ..... + 20²) = 2870, the value of (2² + 4² + 6² + ... + 40² ) is :

A) 2870 B) 5740 C) 11480 D) 28700

Discuss

Answer & Solution

Answer: Option C

Solution:

= (2² +4² +6² + ..... + 40²)
= (1 × 2)² +(2 × 2)² +(2 × 3)² + ..... + (2 × 20)²
= 2² × (1² + 2² + 3² + ..... + 20²)
= (4 × 2870)
= 11480

34

If the number 357*25* is divisible by both 3 and 5, then the missing digits in the unit's place and the thousandth's place respectively are :

A) 0, 6 B) 5, 1 C) 5, 4 D) None of these

Discuss

Answer & Solution

Answer: Option D

Solution:

Let the unit's place be x and the thousand's place be y.
Then, 357y25x is divisible by 5 only when x = 0 or x = 5
Also, this number is divisible by 3 only when sum of its digits is divisible by 3
So, (22 + x + y) must be divisible by 3
∴ x + y = 2
Taking x = 0, we get y = 2
So, the unit place = 0 and thousand's place = 2

35

In the relation x > y + z, x + y > p and z < p, which of the following is necessarily true ?

A) y > p B) x + y > z C) y + p > x D) Insufficient data

Discuss

Answer & Solution

Answer: Option B

Solution:

x + y > p and p > z
⇒ x + y > z

36

(xⁿ - aⁿ) is divisible by (x - a)

A) for all values of n B) only for even values of n C) only for odd values of n D) only for prime values of n

Discuss

Answer & Solution

Answer: Option A

Solution:

We know that (xⁿ - aⁿ ) is always divisible by (x - a) for all values of n.

37

8 + 88 + 888 + 8888 + 88888 + 888888 = ?

A) 897648 B) 896748 C) 986748 D) 987648 E) None of these

Discuss

Answer & Solution

Answer: Option D

Solution:

8 8 8 8 8 8

8 8 8 8 8

8 8 8 8

8 8 8

8 8

+ 8

9 8 7 6 4 8

38

If a + b + c = 0, (a + b) (b + c) (c + a) equals

A) ab (a + b) B) (a + b + c)² C) - abc D) a² + b² + c²

Discuss

Answer & Solution

Answer: Option C

Solution:

a + b + c = 0
⇒ (a + b) = - c
(b + c) = - a
and (c + a) = - b
⇒ (a + b) (b + c) (c + a)
= (- c) × (- a) × (- b)
= - (abc)

39

If x is a rational number and y is an irrational number, then-

A) Both x + y and xy are necessarily rational B) Both x + y and xy are necessarily irrational C) xy is necessarily irrational, but x + y can be either rational or irrational D) x + y is necessarily irrational, but xy can be either rational or irrational

Discuss

Answer & Solution

Answer: Option D

Solution:

(a) Let x = 0 and y = $$\sqrt 2 $$
Then, x is rational and y is irrational.
∴ x + y = 0 + $$\sqrt 2 $$ = $$\sqrt 2 $$ , which is irrational
Thus, x + y is not rational
(b) Let x = 0 and y = $$\sqrt 2 $$
Then, x is rational and y is irrational
∴ xy = 0 × $$\sqrt 2 $$ = 0, which is rational
Hence, xy is not irrational
(c) As shown in (B), xy is not necessarily irrational
(d) x + y is necessary irrational. But xy can be either rational or irrational.
Hence, (D) is true.

40

The digit in the unit's place of [(251)⁹⁸ + (21)²⁹ - (106)¹⁰⁰ + (705)³⁵ - 16⁴ + 259] is :

A) 1 B) 4 C) 5 D) 6

Discuss

Answer & Solution

Answer: Option B

Solution:

Unit digit in [(251)⁹⁸ + (21)²⁹ - (106)¹⁰⁰ + (705)³⁵ - 16⁴ + 259]
= unit digit in (1 + 1 - 6 + 5 - 6 + 9)
= 4