**Notes**
**NCERT Solutions**
**Assignments**
**Revision Notes**
Given below are the Class 10 Maths Problems for Real Numbers

a. HCF and LCM problems

b. Prime Factorisation Problems

c. Division Problems

d. Multiple Choice Problems

e. Word Problems

**Question 1** In a seminar, the number of participants in Hindi, English and Mathematics are 60, 84 and 108, respectively. Find the minimum number of rooms required if in each room the same numbers of participants are to be seated and all of them being in the same subject.

Solution
3
We need to find the HCF of 60,84 and 108

By prime factorisation

60=2*2*3*5

84=2*2*3*7

108=2*2*3*3*3

HCF =12

Rooms required= 60/12 + 84/12+108/12=5+7+9=21

**Question 2** If the HCF of 657 and 963 is expressible in the form $657x + 963 \times (-15)$, find x.

Solution
First Lets find out the HCF of 657 and 963

By prime factorisation

$657 = 9 \times 63$

$963 = 9 \times 107$

HCF(657, 963) = 9

Now it is given that HCF is expressed in the form of $657x + 963 \times (-15)$

Then, $657x + 963 \times (-15) = 9$

$657x - 963 \times 15 = 9$

$657x = 14454$

$x = \frac {14454}{657}$

x = 22

**Question 3** Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

Solution
We have to find the HCF of 285-9 and 1249-7 i.e 276 and 1242
By Euclid division
1242=276X4+138
276=2X138 + 0
So 138 is the HCF

**Question 4** Find the greatest umbers that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.

**Question 5** Find the greatest number which divides 2011 and 2623 leaving remainder 9 and 5 respectively.

**Question 6** Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.

**Question 7** Find the smallest number which when increased by 17 is exactly divisible by both 520 and 468.

**Question 8** A circular field has a circumference of 360km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?

**Question 9** If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF, then the product of two numbers is.

**Question 10** The HCF of two numbers is 145 and their LCM is 2175. If one number is 725, find the other.

**Question 11** If two positive integers

*a *and

*b *are written as

$a = x^3y^2$ and $b = xy^3$ .

*x*,

*y *are prime numbers, then HCF (

*a*,

*b*) is

(A) $xy$

(B) $xy^2$

(C)$x^3y^3$

(D) $x^2y^2$

Solution
HCF is common factor between the number

So HCF=xy^{2 }.Hence Option (B)

**Question 12** If two positive integers

*p *and

*q *can be expressed as

*p *=

*ab*^{2} and

*q *=

*a*^{3}b *a*,

*b *being prime numbers, then LCM (

*p*,

*q*) is

(A)

*ab *
(B)

*a*^{2}b^{2}
(C)

*a*^{3}b^{2}
(D)

*a*^{3}b^{3}
Solution
LCM is Product of the greatest power of each prime factor involved in the number

So LCM=a^{3}b^{2 }.Hence Option (C)

**Question 13** If HCF (26, 169) = 13, then LCM (26, 169) =

**Question 14** If 3 is the least prime factor of number a and 7 is the least prime factor of numbers b, then the least prime factor of a + b, is.

Solution
Since least prime factor of number a =3, It must be odd number

Since least prime factor of number b =7, It must be odd number

Odd Number + Odd number= even number.

Least prime factor of any even number is 2. So Answer is 2

**Question 15** Euclid's Division Lemma states that if p and q are any two positive integers, then there exist unique integers r and s such that

(A) $p = qr + s, 0 \leq s < q$

(B) $p = qr + s, 0 < s \leq q$

(C) $p= qr + s, 0 \leq r < q$

(D) $p = qr + s, 0 < r \leq q$

**Question 16** Two tankers contain 583 litres and 242 litres of petrol respectively. A container with maximum capacity is used which can measure the petrol of either tanker in exact number of litres. How many containers of petrol are there in the first tanker.

(A)53

(B)50

(C)54

(D)11

Other Questions Answer
3. 138

4. 63

5. 154

6. 204

7. 4063

8. 30 days

9. 194400

10. 435

13. 338

15. a

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Class 10 Maths
Class 10 Science