Class 10 Math Exercise 1.1 solution

1. Express each number as product of its prime factors

1. 140  Answer: Prime Factor = 2 × 2 × 5 × 7 = 2² × 5 × 7
2. 156  Answer: Prime Factor = 2 × 2 × 3 × 13 = 2² × 3 × 13
3. 3825  Answer: Prime Factor = 3 × 3 × 5 × 5 × 17 = 3² × 5² × 17
4. 5005  Answer: Prime Factor = 5 × 7 × 11 × 13
5. 7429  Answer: Prime Factor = 17 × 19 × 23

2. Find the LCM and HCF of the pairs and verify LCM × HCF = product

1. (i) 26 and 91
   Work
   26 = 2 × 13
   91 = 7 × 13

   HCF = 13
   LCM = 2 × 7 × 13 = 182

   Verification: 26 × 91 = 13 × 182 = 2366  ✓

2. (ii) 510 and 92
   510 = 2 × 3 × 5 × 17
   92 = 2 × 2 × 23

   HCF = 2
   LCM = 2² × 3 × 5 × 17 × 23 = 23460

   Verification: 510 × 92 = 2 × 23460 = 46920  ✓

3. (iii) 336 and 54
   336 = 2 × 2 × 2 × 2 × 3 × 7 = 2⁴ × 3 × 7
   54 = 2 × 3 × 3 × 3 = 2 × 3³

   HCF = 2 × 3 = 6
   LCM = 2⁴ × 3³ × 7 = 3024

   Verification: 336 × 54 = 6 × 3024 = 18144  ✓

3. LCM and HCF of three integers (prime factor method)

1. (i) 12, 15 and 21
   12 = 2 × 2 × 3 = 2² × 3
   15 = 3 × 5
   21 = 3 × 7

   HCF = 3
   LCM = 2² × 3 × 5 × 7 = 420

2. (ii) 17, 23 and 29
   17 = 17
   23 = 23
   29 = 29

   HCF = 1
   LCM = 17 × 23 × 29 = 11339

3. (iii) 8, 9 and 25
   8 = 2 × 2 × 2 = 2³
   9 = 3 × 3 = 3²
   25 = 5 × 5 = 5²

   HCF = 1
   LCM = 2³ × 3² × 5² = 8 × 9 × 25 = 1800

4. Given HCF(306, 657) = 9, find LCM(306, 657)

Using LCM × HCF = product of the numbers:

LCM(306,657) × 9 = 306 × 657
LCM(306,657) = (306 × 657) / 9 = 34 × 657 = 22338

So, LCM = 22338.

5. Can 6ⁿ end with digit 0 for any natural n?

For a number to end with 0 it must be divisible by 10 = 2 × 5. But 6ⁿ = (2 × 3)ⁿ contains only primes 2 and 3 — no factor 5. Therefore 6ⁿ can never be divisible by 10, so it cannot end with the digit 0 for any natural n.

6. Explain why the given expressions are composite

1. 7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × 78

   Since it factors as a product of integers greater than 1, it is composite.

2. 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 × (7 × 6 × 4 × 3 × 2 × 1 + 1) = 5 × 1009

   Again it’s a product of integers greater than 1, hence composite.

7. Circular path — when will they meet again at the start?

Sonia: 18 minutes per round. Ravi: 12 minutes per round. They start together and move in same direction. They return together after LCM(18,12) minutes.

18 = 2 × 3 × 3 = 2 × 3²
12 = 2 × 2 × 3 = 2² × 3
LCM = 2² × 3² = 4 × 9 = 36

They will meet after 36 minutes.

1.Express each number as product of its prime factors

(i)140 Answer: Prime Factor= 2 × 2 × 5 × 7 = × 5 × 7

(ii)156 Answer: Prime Factor = 2 × 2 × 3 × 13 = × 3 × 13

(iii)3825 Answer:Prime Factor = 3 × 3 × 5 × 5 × 17 = × × 17

(iv)5005 Answer: Prime Factor = 5 × 7 × 11 × 13

(v)7429 Answer: Prime Factor = 17 × 19 × 23

2.Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.(i)26 and 91

Answer:

(i)26 and 91

26 = 2 × 13

91 = 7 × 13

HCF = 13

LCM = 2 × 7 × 13 = 182

Verification,

Product of the two numbers = HCF × LCM (as per rule)

26 × 91 = 13 × 182

2366 = = 2366

Hence,

product of two numbers = HCF × LCM verified

(ii)510 and 92

Answer: (ii) 510 and 92

510 = 2 × 3 × 5 ×17

92 = 2 × 2 × 23

HCF = 2

LCM = × 3 × 5 × 17 × 23

= 4×3×5×17×23=23460

Product of the two numbers = HCF × LCM (………………………….As per rule)

510 × 92 = 2 × 23460

46920 = 46920 Hence, product of two numbers = HCF × LCM verified

(iii) 336 and 54

Answer

(iii) 336 and 54

336 = 2 × 2 × 2 × 2 × 3 × 7 = 2⁴ × 3 × 7

54 = 2 × 3 × 3 × 3 = 2 × 3³

HCF = 2 × 3 = 6

LCM =2⁴ × 3³ × 7 = 24 × 27 × 7 = 3024

Product of the two numbers= HCF × LCM (As per rule)

336 × 54 = 6 × 3024

18144 = 18144

Hence,

product of two numbers = HCF × LCM verified

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21

Answer:

12 = 2 × 2 × 3 = 2²× 3

15 = 3 × 5

21 = 3 × 7

HCF = 3

LCM = 2²× 3 × 5 × 7 = 420

(ii) 17, 23 and 29

Answer:

17 = 1 × 17

23 = 1 × 23

29 = 1 × 29

HCF = 1

LCM = 17 × 23 × 29 = 11339

(iii) 8, 9 and 25

Answer:

(iii) 8, 9 and 25

8 = 2 × 2 × 2

9 = 3 × 3

25 = 5 × 5 =

HCF = 1

LCM = 2³××

= 8 × 9 × 25 = 1800

4. Given that HCF (306, 657) = 9, find LCM (306, 657)

Answer:

We know that,

LCM × HCF = Product of two numbers (……………………………………..as per Rule)

LCM (306,657)x HCF(306,657) = 306 x 657

LCM (306,657) x 9 = 306 x 657

LCM (306, 657) =

=34×657=22338

SO,

LCM OF 306 AND 657 =22338

5. Check whether 6ⁿ can end with the digit 0 for any natural number n.

Answer:

For a number to end with the digit 0, it must be divisible by 10, which means it has to include both 2 and 5 as factors (LIKE 10=2×5 ).

Now,

the prime factorisation of 6^𝑛 as (2 ∗ 3)^𝑛 This expression only contains the primes 2 and 3 — there is no 5 present in the factorisation.

Because of this, 6^𝑛

can never be divisible by 5, and therefore it can never be divisible by 10.

So, for any natural number will never end with the digit 0.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Answer:

We know that ,

composite numbers have factors other than 1 and itself for (example 4=1,2 and 4 or 6=1,2,3 and 6)

Let’s look at the first expression:

7×11×13+13

Factorising, we get:

=13×(7×11+1)

=13×(77+1)

=13×78

So this number can be written as a product of 13 and 78, it clearly has factors other than 1 and itself. That means it is a

composite number

Second part 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5

= 5 ×(7 × 6 × 4 × 3 × 2 × 1 + 1)

= 5 × (1008 + 1) = 5 ×1009

1009 cannot be factorised further. Therefore, the given expression has 5 and 1009 as its factors. Hence, it is a composite number.

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Answer:

They’ll meet at the starting point after the LCM of 18 and 12 minutes. 18=2×3×3=2×3²

12=2×2×3=2²×3

LCM of 12 and 18=3² × 2²

=9×4=36

So, they meet after 36 minutes.