# Number System MCQ's

1) Every rational number is a -

1. Whole number
2. Real number
3. Natural number
4. None of the above

Explanation: The numbers which can be found on the number line and include both rational and irrational numbers are known as real numbers, e.g., -1.5,√2,0,1,2,3,π.Almost any number which you can imagine is a real number.

2) Between any two numbers, there are -

1. Two rational numbers
2. No rational number
3. Infinite rational numbers
4. None of the above

Explanation: There are infinite rational numbers in between any two numbers.

3) What will be the value of x3 + y3 + z3, if x + y + z = 0?

1. 3xyz
2. 2xyz
3. xyz
4. xyz(xy + yz + zx)

Explanation: It is given that

x + y + z = 0

On cubing both sides, we will get

(x + y + z)3 = 0

x3 + y3 + z3 - 3xyz = 0

So, the value of x3 + y3 + z3 = 3xyz

4) Digit 1 is occurring 136 times on writing all of the page numbers of a book. What will be the number of pages in the book?

1. 194
2. 195
3. 200
4. 295

Explanation: From 1 - 99, the digit 1 occurs 20 times, and from 100 - 199, the digit 1 occurs 120 times.

So, from 1 to 199, the digit 1 occurs -

20 + 120 = 140 times

According to question 1 is occurring only 136 times, which means we need to remove 196, 197, 198, and 199. So, the required number of pages will be 195.

5) Which of the following is the unit digit in the product of 853 x 452 x 226 x 1346?

1. 2
2. 5
3. 6
4. 7

Explanation: Pick up the unit digit of each number and multiply them;

3 in 853

2 in 452

6 in 226

6 in 1346

∴ 3 x 6 x 2 x 6 = 216 (consider the unit digit in the product)

So, the unit digit in the product of 853 * 1346 * 452 * 226 is 6.

6) The sum of odd numbers upto 240 is -

1. 11400
2. 12400
3. 13400
4. 14400

Explanation: Number of odd numbers up to 240 = 240/2 = 120

Sum of first n odd numbers = n2

n = 120

Required sum = 1202 = 14400

7) Which of the following number is divisible by 9?

1. 56785
2. 45678
3. 65889
4. 67578

Explanation: A number is divisible by 9 if the sum of its digits is divisible by 9.

The Sum of the digits of the given numbers are;

6+7+5+7+8= 33

5+6+7+8+5= 31

4+5+6+7+8= 30

6+5+8+8+9= 36

The sum of the digits of the number 65889 is 36, which is divisible by 9, so the correct answer is 65889.

8) What smallest number should be subtracted from 9805 so that it is divisible by 8?

1. 6
2. 7
3. 5
4. 8

Explanation: On dividing 9805 by 8, the remainder is 5. So, 5 is the smallest number which should be subtracted from 9805 to make it divisible by 8.

9) Which of the following is the unit digit in the product of {(341)491 x (625)317 x (6374)1793}?

1. 3
2. 7
3. 0
4. 8

Explanation: Pick up the unit digit of each number and multiply them;

Unit digit in (341)491

= Unit digit in (1)491 = 1

Unit digit in (625)317

= Unit digit in (5)317 = 5

Unit digit in (6374)1793

= (4)1793

= unit digit in [(42)896 x 4]

= unit digit in 6 x 4 = 4

So, on multiplying the unit digits, we will get -

1 x 5 x 4 = 20 (consider the unit digit in the product)

So, the unit digit in the product of {(341)491 x (625)317 x (6374)1793} is 0.

10) Which of the following is completely divisible by 45?

1. 331145
2. 306990
3. 181660
4. None of the above

Explanation: A number that can be divisible by 3, 5, and 9 is also divisible by 45.

So, we check the divisibility of given numbers by the following rules -

A number is divisible by 3 if the sum of its entire digits is divisible by 3.

The number which ends with 0 or 5 is divisible by 5.

A number is divisible by 9 if the sum of its entire digits is divisible by 9.

The number 306990 fulfills all the requirements, so the answer is 306990.

11) If the two-third of three - fourth of a number is 34, what will be the 20% of that number?

1. 13.4
2. 13.6
3. 13.7
4. 14

Explanation: Let the number be X.

According to the question,

2/3 * 3/4 * X = 34

6/12 * X = 34

Or,

1/2 * X = 34

So, X = 68

Now, 20% of 68 is = 68 * 20/100 = 13.6

12) 7X2 is a three-digit number in which X is a missing digit. If the number is divisible by 6, the missing digit is -

1. 4
2. 3
3. 7
4. 5

Explanation: The given number is divisible by 6, so it would be divisible by 2 and 3. As the last digit is 2, whatever be the value of X, it would be divisible by 2.

Now, 7 + X + 2 = 9 + X, must be divisible by 3.

So, X = 3 makes the number divisible by 3, so 3 is the required digit.

13) Which is the largest 4-digit number that can be exactly divisible by 66?

1. 9987
2. 9912
3. 9913
4. 9966

Explanation: The largest four-digit number is = 9999, and on dividing it with 66, we will get 33 as the remainder.

So, the largest 4-digit number divided by 66 = 9999 - 33 = 9966

14) Which is the largest 4-digit number that can be exactly divisible by 66?

1. 9987
2. 9912
3. 9913
4. 9966

Explanation: The largest four-digit number is = 9999, and on dividing it with 66, we will get 33 as the remainder.

So, the largest 4-digit number divided by 66 = 9999 - 33 = 9966

15) What will be the remainder when 636 is divided by 215?

1. 3
2. 2
3. 1
4. None of the above

Explanation: We can write 636/215 as

(63)12/215

Or, we can say 21612/215

We will always get remainder 1 on dividing 216 by 215

= 112/215

So, the remainder will be 1.

16) Which of the following is the least number which will leave the remainder 5, when divided by 8, 12, 16, and 20?

1. 245
2. 255
3. 265
4. None of the above

Explanation: First we need to find the least number, so we have to find out the LCM of 8, 12, 16, and 20.

8 = 2 x 2 x 2

12 = 2 x 2 x 3

16 = 2 x 2 x 2 x 2

20 = 2 x 2 x 5

LCM = 2 x 2 x 2 x 2 x 3 x 5 = 240

240 is the least number that is exactly divisible by 8, 12, 16, and 20.

So, the required number that will leave remainder 5 is -

240 + 5 = 245

17) Which of the following is largest among others?

1. √0.0004
2. √0.0121
3. (0.1)2
4. 0.12

Explanation: Let's examine each option individually,

√0.0004 = 0.02

√0.0121 = 0.11

(0.1)2 = 0.1

0.12 = 0.12

From the above examination, we can see 0.12 is the largest among others.

18) What will be the unit digit of (2153)167?

1. 1
2. 7
3. 3
4. 9

Explanation: We have 2153167, and the unit digit is = 3167

As we know,

31 = 3 = unit digit is 3

32 = 9 = unit digit is 9

33 = 27 = unit digit is 7

34 = 81 = unit digit is 1

The cycle will continue. On the dividing the power of 3 by 4, we will get -

167/4 = remainder is 3

So, 33 gives us the unit digit 7. So, the unit digit of 2153167 is 7.

19) If the sum of two numbers is considered as 'a' and their product is considered as 'b', then what will be the sum of their reciprocals?

1. a/b
2. 1/b + 1/b
3. b/a
4. ab

Explanation: Suppose the numbers are P and Q

So, according to the question -

P + Q = a

P * Q = b

Sum of reciprocals of P and Q is = 1/P + 1/Q

= Q + P/PQ

= a/b

20) From the list of below options, which of the fraction is the smallest?

1. 14/33
2. 7/13
3. 11/13
4. 8/15

Explanation: Suppose the numbers are P and Q

So, according to the question -

P + Q = a

P * Q = b

Sum of reciprocals of P and Q is = 1/P + 1/Q

= Q + P/PQ

= a/b

21) If the number A381 is divisible by 11, then what is the value of A?

1. 7
2. 3
3. 1
4. 8

Explanation: A number is divisible by 11 if the difference of the sum of digits on odd places and the sum of digits on even places is zero or divisible by 11. So, we have the number A381.

Hence, (A + 8) - (3 + 1) = either 0 or multiple of 11

If we put A = 7, then we get

(7 + 8) - (3 + 1) = 15 - 4 = 11 (Which is divisible by 11).

22) Suppose a number 381A is divisible by 9, then what is the value of A?

1. 6
2. 3
3. 1
4. 8

Explanation: A number is divisible by 9 if the sum of its digits is divisible by 9, e.g., 117. So, we have the number 381A.

Hence, 3 + 8 + 1 + A = multiple of 9

If we put A = 6, then we get

3 + 8 + 1 + 6 = 18 (which is a multiple of 9)

23) Suppose there is a number 'n'. When 'n' is divided by 5, the remainder will be 2. What will be the remainder when n2 is divided by 5?

1. 6
2. 4
3. 1
4. 8

Explanation: According to the question when 'n' is divided by 5, we will get the remainder 2

n/5 = remainder 2

On putting n =7, and on dividing it with 5 we will get remainder 2

So, n = 7, and n2 = 49

n2/5 = 49/5 = remainder 4

24) If the difference between three times and seven times of a number is equal to 36, what will be the number?

1. 9
2. 4
3. 1
4. 8

Explanation: Let the number be X.

So according to the question -

7X - 3X = 36

4X = 36

Or,

X = 36/4 = 9

So, the required number is 9.

25) What will be the value of x, if 5(x + 3) = 25(3x - 4)?

1. 11/5
2. 11/6
3. 5/11
4. 6/11

Explanation: Given, 5(x + 3) = 25(3x - 4)

We can write it as -

5(x + 3) = 52 x (3x - 4)

Or,

x + 3 = 2(3x - 4)

x + 3 = 6x - 8

or, 5x = 11

So, x = 11/5

26) If the sum of two numbers is 12 and their product is 35, then what will be the sum of their reciprocals?

1. 12/35
2. 1/5
3. 2/3
4. None of the above

Explanation: Suppose the numbers are P and Q

So, according to the question -

P + Q = 12

P * Q = 35

Sum of reciprocals of P and Q is = 1/P + 1/Q

= Q + P/PQ

= 12/35

27) What will be the value of a3 - 3a2 + 3a + 3b + 3b2 + b3, if a = -4, and b = -2?

1. 126
2. -125
3. -126
4. 125

Explanation: Given a = -4 and b = -2

On putting the values in the given equation, we will get -

(-4)3 - 3 x (-4)2 + 3 x (-4) + 3 x (-2) + 3 x (-2)2 + (-2)3

= -64 - 48 - 12 - 6 + 12 - 8

= -126

28) If the sum and reciprocal of a number is equal to 2, the number is -

1. 1
2. -1
3. -2
4. 2

Explanation: Suppose the number is a.

So, according to the question -

a + 1/a = 2

(a2 + 1)/a = 2

a2 + 1 = 2a

Or, a2 + 1 -2a = 0

We can say that (a - 1)2 = 0

So, a =1

Hence the required number is 1, whose sum and reciprocal are equal to 2.

29) If the ratio of two positive numbers is 7 : 9 and their product is 1575, then the greatest number is -

1. 45
2. 15
3. 35
4. 55

Explanation: Given ration 7 : 9

So, let the numbers be 7a and 9a. According to the question -

7a * 9a = 1575

63a2 = 1575

a2 = 1575/63

Or a2 = 25

And a =5

So, the two numbers will be -

7a = 7 * 5 = 35

And 9a = 9 * 5 = 45

Hence, the greater number is 45 between the given numbers.

30) The sum of two numbers is equal to the thrice of their difference. What will be the ratio between them?

1. 1 : 3
2. 3 : 1
3. 1 : 2
4. 2 : 1

Explanation: Let the numbers be 'a' and 'b'. According to the question -

(a + b) = 3(a - b)

(a + b) = 3a - 3b

2a = 4b

a = 2b

So, a/b = 2 : 1

31) Which of the following is equal to x3?

1. x6x3
2. x6 + x3
3. x6 / x3
4. (x6)3

Explanation: We can write x6 / x3 as,

x6 - 3 = x3

32) The sum of four consecutive even numbers is 748, then which is the smallest number among them?

1. 184
2. 186
3. 174
4. None of the above

Explanation: Let a be the smallest even number

So, according to the question -

(a) + (a + 2) + (a + 4) + (a + 6) = 748

4(a) + 12 = 748

4a = 736

Or, a = 184

So, the smallest number be 184

Alternative solution

We can use an alternative solution of it

Middle term = 748/4 = 187

So, the numbers will 184, 186, 188, and 190

And the smallest number between them is 184.

33) If (64)2 - (36)2 = 20 * x, then what is the value of x?

1. 120
2. 140
3. 112
4. None of the above

Explanation: Here, we can apply the formula of a2 - b2 = (a + b) (a - b)

So, after applying the formula, we will get

(64 + 36) (64 - 36) = 20 * x

100 * 28 = 20 * x

x = 100 * 28/ 20 = 140

34) What is the value of (a - b), if (a2 - b2)/(a + b) = 25?

1. 36
2. 22
3. 45
4. 25

Explanation: According to the identity,

a2 - b2 = (a + b) (a - b)

It is given that (a2 - b2)/(a + b) = 25

So, it can be written as (a + b) (a - b)/(a + b) = 25

Hence, (a - b) = 25

35) What will be the result of 1397 x 1397?

1. 1951609
2. 1951608
3. 1951699
4. None of the above

Explanation: Given 1397 x 1397

We can also write it as (1397)2

Or, (1400 - 3)2

On expanding we get,

= (1400)2 + (3)2 - (2 x 1400 x 3)

= 1960000 + 9 - 8400

= 1960009 - 8400

= 1951609

36) Which of the following is the unit digit in (1570)2 + (1571)2 + (1572)2 + (1573)2?

1. 2
2. 4
3. 6
4. 7

Explanation: Pick up the unit digit of each number and multiply them;

02 in (1570)2 = 0

12 in (1571)2 = 1

22 in (1572)2 = 4

32 in (1573)2 = 9

On adding the unit digits, we will get,

0 + 1 + 4 + 9 = 14 (unit digit = 4)

So, on adding (1570)2 + (1571)2 + (1572)2 + (1573)2, the unit digit will be 4.

37) Suppose x = a(b - c), y = b(c - a), z = c(a - b), then what is the value of (x/a)3 + (y/b)3 + (z/c)3?

1. 3xyz/abc
2. 2xyz/abc
3. xyz/abc
4. None of the above

Explanation: Given x = a(b - c), y = b(c - a), z = c(a - b)

x = a(b - c)

or, x/a = (b - c) …..(i)

y = b(c - a)

or, y/b = (c - a) …..(ii)

z = c(a - b)

or, z/c = (a - b) …..(iii)

Add equation (i), (ii), and (iii), we will get

x/a + y/b + z/c = b - c + c - a + a - b

x/a + y/b + z/c = 0

So, (x/a)3 + (y/b)3 + (z/c)3 = 3 (x/a) * (y/b) * (z/c) [because if a + b + c = 0, then a3 + b3 + c3 = 3abc]

= 3xyz/abc

38) The sum of three numbers is 2, if the first number is ½ times of the 2nd number, and the third number is ¼ times of the 2nd number. What will be the second number?

1. 8/7
2. 7/8
3. 6/7
4. 7/6

Explanation: Suppose the numbers be a, b, and c. So, according to the question,

a + b + c = 2

and, a = ½b ….(i)

c = ¼b ….(ii)

From equation (i),

a/b = ½

and from equation (ii)

b/c = 4/1

Ratio between a, b, c is

a : b : c = 2x : 4x : 1x

(2x + 4x + 1x) = 2

7x = 2

So, x = 2/7

Then a = 2x = 2 * 2/7 = 4/7

b = 4x = 4 * 2/7 = 8/7

c = x = 2/7

So, the second number is 8/7.

39) The sum of the squares of three consecutive positive numbers is 365. What will the sum of numbers?

1. 36
2. 33
3. 45
4. None of the above

Explanation: Suppose the three consecutive positive numbers be x, x + 1, and x + 2

According to the question,

(x)2 + (x + 1)2 + (x + 2)2 = 365

On expanding, we will get,

x2 + x2 + 1 + 2x + x2 + 4 + 4x = 365

3x2 + 6x = 360

Or, x2 + 2x - 120 = 0

=> (x - 10) (x + 12) = 0

So, x = 10

First number x = 10

Second number x + 1 = 11

Third number x + 2 = 12

So, sum of the numbers is = 10 + 11 + 12 = 33

40) The sum of three numbers is 252. What is the value of the second number if the first number is thrice of the second number, and the third number is two-third of the first number?

1. 36
2. 42
3. 45
4. None of the above

Explanation: Suppose the three numbers be a, b, and c

According to the question,

a = 3b or, a/b = 3/1

c = 2/3a or, a/c = 3/2

Ratio between a, b, c is

a : b : c = 3x : x : 2x

(3x + x + 2x) = 252

6x = 252

So, x = 252/6 = 42

Then, first number a = 3x = 3 * 42 = 126

Second number, b = x = 42

Third number, c = 2x = 42 * 2 = 84

So, the second number is 42.

41) If x, y, and z are said to be the real numbers, then what is the value of (x - y)3 + (y - z)3 + (z - x)3 ?

1. 0
2. (x - y)
3. 3(x - y) (y - z) (z - x)
4. (x - y) (y - z) (z - x)

Answer: (c) 3(x - y) (y - z) (z - x)

Explanation: Suppose a = (x - y), b = (y - z), and c = (z - x)

On adding a, b, and c we will get

a + b + c = x -y + y - z + z - x

=> a + b + c =0

So, a3 + b3 + c3 = 3abc [because if a + b + c = 0, then a3 + b3 + c3 = 3abc]

We can say that (x - y)3 + (y - z)3 + (z - x)3 = 3 (x - y) (y - z) (z - x)

42) If the sum of the digits of a two-digit number is 12, and the difference between the digits of that number is 6, what will be the number?

1. 36 or 63
2. 22 or 22
3. 39 or 93
4. 25 or 52

Explanation: We can solve it via two ways. Let's see the ways.

Shortcut method:

We can also get the answer using the specified options. We can check the digits of the numbers mentioned in the options. As in the option 'C', there are 39 or 93, and on adding their digits, we will get 12, and on subtracting the digits, we will get 6.

Detailed method:

Suppose the digits of the two-digit number be 'a' and 'b'.

So, according to the question

a + b = 12 ….(i)

a - b = 6 ……(ii)

Add equation (i) and (ii), we get

2a = 18

Or, a = 9

Put a = 9 in eqn (i), we will get

b = 3

So, the two-digit number is 93 (when a > b)

Or 39 (when b > a)

43) If the three-fifth of a number is equal to 70% of another number, what is the ratio between the first number and second number?

1. 6 : 7
2. 7 : 6
3. 3 : 4
4. 4 : 3

Explanation: Let the numbers be a and b.

So, according to the question,

a * 3/5 = 70% of b

3a/5 = 70b/100

a/b = (70 * 5)/(100 * 3) = 7/6

So, the ration between a and b is 7 : 6.

44) If the sum of the cubes of three numbers is 4500, and their ratio is 1 : 2 : 3, what is the value of the smallest number between them?

1. 6
2. 5
3. 3
4. 4

Explanation: Given ratio is 1 : 2 : 3

So, the numbers will be 1a : 2a : 3a

Then, a3 + 8a3 + 27a3 = 4500

36a3 = 4500

Or, a3 = 4500/36 =125

So, a = 5

The smallest number is 5.

45) The denominator of a fraction is 3 more than its numerator. If the denominator is decreased by 2, and the numerator is increased by 7, we will get 2. What will be the sum of the numerator and denominator of that fraction?

1. 6
2. 13
3. 31
4. 14

Explanation: Let the numerator be 'a' so, denominator will be 'a + 3'.

According to the question,

(a + 7)/ ((a + 3) - 2) = 2/1

(a + 7)/(a + 1) = 2/1

2a + 2 = a + 7

=> a = 5

So, the fraction is a/a + 3 = 5/ 5 + 3

= 5/8

And the sum of the numerator and denominator of the fraction is 13.

46) Which of the following is the result of 1505 x 1505?

1. 2265015
2. 2265035
3. 2265025
4. None of the above

Explanation: Given: 1505 x 1505

= 15052

= (1500 + 5)2

Apply formula; (a + b)2 = a2 + b2 + 2ab

= 15002 + 52 + 2 *1500* 5

= 2250000 + 25 + 15000

= 2265025

47) A boy has mistakenly multiplied a number by 45 instead of multiplying it with 25. Due to this, the answer was 200 more than the correct answer. What was the number?

1. 15
2. 10
3. 7
4. None of the above

Explanation: The required number = Increase in result/(wrong multiplier - correct multiplier)

= 200/45 - 25

= 10

48) If a = (0.4)2, b = 0.04, and c = 2/5, then which of the following is the correct relationship between a, b, and c?

1. c > a > b
2. a > b > c
3. a < b < c
4. None of the above

Answer: (a) c > a > b

Explanation: a = (0.4)2 = 0.16

b = 0.04

c = 2/5 = 0.4

So, the relationship is c > a > b.

49) The sum of the numerator and denominator of a fraction is 11. If we add 2 to both numerator and denominator, the fraction will be increased by 1/24. What is the difference between the numerator and denominator of that fraction?

1. 1
2. 2
3. 6
4. 8

Explanation: Let the numerator of the fraction be x, so the denominator will be 11 - x

So, the fraction = x/11 - x

On adding 2 in both numerator and denominator, according to the question, the fraction will be -

(x + 2)/(11 - x + 2) = (x)/(11 - x) + 1/24

(x + 2)/(13 - x) - (x)/(11 - x) = 1/24

[11x + 22 - x2 -2x - 13x + x2]/(13 - x) (11 - x) = 1/24

After solving the above equation, we will get

=> 528 - 96x = 143 - 24x + x2

x2 + 72x - 385 = 0

(x + 77) (x - 5) = 0

So, x =5

So, numerator = 5

and, denominator = 11 - 5 = 6

Difference between both is = 6 - 5 = 1

50) If a and b are two odd numbers, then which of the following is even?

1. a + b + 2ab
2. a + b + ab
3. a + b + 1
4. None of the above

Answer: (a) a + b + 2ab

Explanation: None

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