Real numbers class 10 questions with solutions

Q 1 :

If two positive integers p and q can be expressed as $p = a b^{2}$ and $q = a^{3} b;$ $a, b$ being prime numbers, then LCM (p, q) is

$a b$
$a^{2} b^{2}$
$a^{3} b^{2}$
$a^{3} b^{3}$

1

2

3

4

(3) $a^{3} b^{2}$

Q 2 :

Let a and b be two positive integers such that $a = p^{3} q^{4} and b = p^{2} q^{3}$ , where p and q are prime numbers. If $HCF (a, b) = p^{m} q^{n}$ and $LCM (a, b) = p^{r} q^{s},$ then $(m + n) (r + s) =$

15
30
35
72

1

2

3

4

(3)

$a = p^{3} q^{4} and b = p^{2} q^{3}$

$HCF (a, b) = p^{2} q^{3}$ ...(i)

and $LCM (a, b) = p^{3} q^{4}$ ...(ii)

But gives: $HCF (a, b) = p^{m} q^{n}$ and $LCM (a, b) = p^{r} q^{s}$

From eq. (i), $p^{m} q^{n} = p^{2} q^{3}$

So, $m = 2$ and $n = 3$

From eq. (ii), $p^{r} q^{s} = p^{3} q^{4}$

So, $r = 3$ and $s = 4$

$∴$ $(m + n) (r + s) = (2 + 3) (3 + 4) = 35 .$

Q 3 :

If two positive integers p and q can be expressed as $p = 18 a^{2} b^{4} and q = 20 a^{3} b^{2},$ where a and b are prime numbers, then LCM (p, q) is :

$2 a^{2} b^{2}$
$180 a^{2} b^{2}$
$12 a^{2} b^{2}$
$180 a^{3} b^{4}$

1

2

3

4

(4)

Given, $p = 18 a^{2} b^{4}$ and $q = 20 a^{3} b^{2}$

$LCM (p, q) = LCM (18 a^{2} b^{4}, 20 a^{3} b^{2}) = 180 a^{3} b^{4}$

Q 4 :

Two positive integers m and n are expressed as $m = p^{5} q^{2}$ and $n = p^{3} q^{4}$ , where p and q are prime numbers. The LCM of m and n is :

$p^{8} q^{6}$
$p^{3} q^{2}$
$p^{5} q^{4}$
$p^{5} q^{2} + p^{3} q^{4}$

1

2

3

4

(3) $p^{5} q^{4}$

Q 5 :

If the HCF(2520, 6600) = 40 and LCM(2520, 6600) = 252 × k, then the value of k is

1650
1600
165
1625

1

2

3

4

(1)

Given, HCF = 40 and LCM = 252 × k

We know that, LCM × HCF = Product of two number

$\Rightarrow 40 \times 252 \times k = 2520 \times 6600$

$\Rightarrow k = \frac{2520 \times 6600}{40 \times 252}$

$\Rightarrow k = 1650$

Q 6 :

If the prime factorisation of 2520 is $2^{3} \times 3^{a} \times 𝑏 \times 7$ , then the value of a + 2b is:

12
10
9
7

1

2

3

4

(1)

2520 = 2 × 2 × 2 × 3 × 3 × 5 × 7
$= 2 ³ \times 3 ² \times 5 \times 7$
$C o m p a r i n g w i t h 2520 = 2 ³ \times 3 ᵃ \times b \times 7$
$W e g e t, a = 2, b = 5 ∴ a + 2 b = 2 + 2 \times 5 = 2 + 10 = 12$

Q 7 :

The LCM of the smallest prime number and the smallest odd composite number is:

10
6
9
18

1

2

3

4

(4)

We have, the smallest prime number = 2

and the smallest odd composite number = 9

∴ LCM of the smallest prime number and the smallest odd composite number

= LCM (2, 9) = 18

Q 8 :

If a = $2 ² \times 3 ˣ, b = 2 ² \times 3 \times 5, c = 2 ² \times 3 \times 7$ and LCM (a, b, c) = 3780, then x is equal to:

1
2
3
0

1

2

3

4

(3)

If LCM (a,b,c) = 3780
By prime factorisation of 3780

$3780 = 2^{2} \times 3^{3} \times 5 \times 7 L C M (a, b, c) = 2^{2} \times 3^{3} \times 5 \times 7 (i) a = 2^{2} \times 3^{x}; b = 2^{2} \times 3 \times 5; c = 2^{2} \times 3 \times 7 L C M (a, b, c) = 2^{2} \times 3^{x} \times 5 \times 7 (i i) C o m p a r i n g (i) a n d (i i), w e g e t 𝑥 = 3$

Q 9 :

The LCM of smallest 2-digit number and smallest composite number is:

12
4
20
40

1

2

3

4

(3)

Smallest 2-digit number = 10 and smallest composite number = 4
LCM of 10 and 4 = 20

Q 10 :

The total number of factors of a prime number is:

1
0
2
3

1

2

3

4

(3)

A prime number has only 2 factors, i.e. the number itself and 1.

Topic Question Set

Q 1 :

If two positive integers p and q can be expressed as $p = a b^{2}$ and $q = a^{3} b;$ $a, b$ being prime numbers, then LCM (p, q) is

Q 2 :

Let a and b be two positive integers such that $a = p^{3} q^{4} and b = p^{2} q^{3}$ , where p and q are prime numbers. If $HCF (a, b) = p^{m} q^{n}$ and $LCM (a, b) = p^{r} q^{s},$ then $(m + n) (r + s) =$

Q 3 :

If two positive integers p and q can be expressed as $p = 18 a^{2} b^{4} and q = 20 a^{3} b^{2},$ where a and b are prime numbers, then LCM (p, q) is :

Q 4 :

Two positive integers m and n are expressed as $m = p^{5} q^{2}$ and $n = p^{3} q^{4}$ , where p and q are prime numbers. The LCM of m and n is :

Q 5 :

If the HCF(2520, 6600) = 40 and LCM(2520, 6600) = 252 × k, then the value of k is

Q 6 :

If the prime factorisation of 2520 is $2^{3} \times 3^{a} \times 𝑏 \times 7$ , then the value of a + 2b is:

Q 7 :

The LCM of the smallest prime number and the smallest odd composite number is:

Q 8 :

If a = $2 ² \times 3 ˣ, b = 2 ² \times 3 \times 5, c = 2 ² \times 3 \times 7$ and LCM (a, b, c) = 3780, then x is equal to:

Q 9 :

The LCM of smallest 2-digit number and smallest composite number is:

Q 10 :

The total number of factors of a prime number is:

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Topic Question Set

Q 7 : The LCM of the smallest prime number and the smallest odd composite number is: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 9 : The LCM of smallest 2-digit number and smallest composite number is: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 10 : The total number of factors of a prime number is: .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

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Q 7 :

The LCM of the smallest prime number and the smallest odd composite number is:

Q 9 :

The LCM of smallest 2-digit number and smallest composite number is:

Q 10 :

The total number of factors of a prime number is: