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This mock test of Real Numbers - Olympiad Level MCQ, Class 10 Mathematics for Class 10 helps you for every Class 10 entrance exam.
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QUESTION: 1

**Q. **The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is:

Solution:

Required number = H.C.F. of (1657 - 6) and (2037 - 5)

= H.C.F. of 1651 and 2032 = 127.

QUESTION: 2

The largest possible number with which when 60 and 98 are divided, leaves the remainder 3 in each case, is –

Solution:

QUESTION: 3

The largest possible number with which when 38, 66 and 80 are divided,leaves the remainder same is –

Solution:

Trick is HCF of (237-132), (132-62), (237-62)

= HCF of (70,105,175) = 35

QUESTION: 4

What is the least possible number which when divided by 24, 32 or 42 in each case it leaves the remainder 5?

Solution:

LCM of 24 , 32 & 42 = 672

The least possible number for your given case = 672 +5 = 677

QUESTION: 5

What is the least possible number which when divided by 24, 32 or 42 in each case it leaves the remainder 5.how many numbers are possible between 666 and 8888?

Solution:

QUESTION: 6

The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is:

Solution:

L.C.M. of 5, 6, 7, 8 = 840.

Required number is of the form 840*k + 3*

Least value of *k* for which (840*k* + 3) is divisible by 9 is *k* = 2.

Required number = (840 x 2 + 3) = 1683.

QUESTION: 7

What is the least possible number which when divided by 18, 35 or 42 leaves 2, 19, 26 as the remainders

respectively?

Solution:

QUESTION: 8

What is the least possible number which when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4, 5 respectively?

Solution:

Here,

2 - 1 = 1

3 - 2 = 1

4 - 3 = 1

5 - 4 = 1

6 - 5 = 1

That is, if we subtract 1 from the number divisible by 2, 3, 4, 5, 6, we get the required number.

The smallest number divisible by 2, 3, 4, 5, 6 will be the LCM of 2, 3, 4, 5, 6.

LCM of 2,3, 4, 5, 6 is 60

So, the required number which leaves a remainder of 1,2,3,4,5= L.C.M of (2,3,4,5,6)-1=59

QUESTION: 9

What is the least possible number which when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4, 5 respectively.what is the least possible 3 digit number which is divisible by 11?

Solution:

QUESTION: 10

How many numbers lie between 11 and 1111 which when divided by 9 leave a remainder of 6 and when divided

by 21 leave a remainder of 12?

Solution:

Number lies between 11 and 1111 When divide by 9 a remainder of 6 and when divided by 21 leave a remainder of 12 = 33, 96, 159 (all separated by 63)

a =33 d=63

1111=a+(n-1)63

1111=33+(n-1)63

1078=(n-1)63

1078/63=n-1

n=(1078+63)/63

n= 1141/63

n = 18.11

then total number=18

QUESTION: 11

The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is

Solution:

Since, 5 and 8 are the remainders of 70 and 125, respectively. Thus, after subtracting these remainders from the numbers, we have the numbers 65 = (70-5),

117 = (125 – 8), which is divisible by the required number.

Now, required number = HCF of 65,117 [for the largest number]

QUESTION: 12

If x|y, where x > 0, y > 0 (x, y ε z) then –

Solution:

QUESTION: 13

If a|b, then gcd of a and b is –

Solution:

QUESTION: 14

A number 10x + y is multiplied by another number 10a + b and the result comes as 100p + 10q +r, where r = 2y, q = 2(x + y) and p = 2x; x, y < 5, q ≠ 0. The value of 10a + b may be:

Solution:

According to problem

(10x+y)(10a+b)=100p + 10q + r

By substituting the values

r=2y

q=(2x+y)

p=2x

(10x+y)(10a+b)=100*2x+10*2(x+y)+2y

(10x+y)(10a+b)=220x+22y

10a+b=22

QUESTION: 15

If x, y ε R and |x| + |y|= 0, then –

Solution:

QUESTION: 16

If a, b, c ε R and a^{2} + b^{2} + c^{2} = ab + bc + ca, then –

Solution:

Consider,

a^{2} + b^{2} + c^{2} – ab – bc – ca = 0

Multiply both sides with 2, we get

2( a^{2} + b^{2} + c^{2} – ab – bc – ca) = 0

⇒ 2a^{2 }+ 2b^{2} + 2c^{2} – 2ab – 2bc – 2ca = 0

⇒ a^{2 }+ a^{2 }+ b^{2} + b^{2 }+ c^{2} + c^{2}– 2ab – 2bc – 2ca = 0

⇒ (a^{2} – 2ab + b^{2}) + (b^{2} – 2bc + c^{2}) + (c^{2} – 2ca + a^{2}) = 0

⇒ (a – b)^{2} + (b – c)^{2} + (c – a)^{2} = 0

Since the sum of squares is zero, it means each term should be zero.

⇒ (a – b)^{2} = 0, (b – c)^{2} = 0, (c – a)^{2} = 0

⇒ a – b = 0, b – c = 0, c – a = 0

⇒ a = b, b = c, c = a

⇒ a = b = c

QUESTION: 17

If x, y ε R and x < y ⇒ x^{2} > y^{2} then –

Solution:

QUESTION: 18

If x, y ε R and x > y ⇒ |x| > |y|, then–

Solution:

QUESTION: 19

If x, y ε R and x > y ⇒ |x| < |y|, then–

Solution:

QUESTION: 20

π and e are –

Solution:

QUESTION: 21

If a, b ε R and a < b, then

Solution:

QUESTION: 22

If x is a non-zero rational number and xy is irrational, then y must be –

Solution:

QUESTION: 23

The arithmetical fraction that exceeds it's square by the greatest quantity is –

Solution:

QUESTION: 24

If x and y are rational numbers such that is irrational, then is

Solution:

QUESTION: 25

If x and y are positive real numbers, then –

Solution:

QUESTION: 26

If two positive integers a and b are written as a = x^{3}y^{2} and b = xy^{3}; x, y are prime numbers, then HCF (a, b) is

Solution:

QUESTION: 27

If x ε R, then |x|=

Solution:

QUESTION: 28

is equal to –

Solution:

QUESTION: 29

+ is equal to –

Solution:

QUESTION: 30

The expression is equal to –

Solution:

QUESTION: 31

If x, y, z are real numbers such that then the values of x, y, z are respectively

Solution:

QUESTION: 32

if a, b, c ε R and a > b ⇒ ac < bc, then –

Solution:

QUESTION: 33

If a, b, c ε R and ac = bc ⇒ a = b, then –

Solution:

QUESTION: 34

Between any two distinct rational numbers –

Solution:

QUESTION: 35

The total number of divisors of 10500 except 1 and itself is –

Solution:

QUESTION: 36

The sum of the factors of 19600 is –

Solution:

QUESTION: 37

The product of divisors of 7056 is –

Solution:

QUESTION: 38

The number of odd factors (or divisors) of 24 is –

Solution:

QUESTION: 39

The number of even factors (or divisors) of 24 is –

Solution:

QUESTION: 40

In how many ways can 576 be expressed as a product of two distinct factors?

Solution:

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