Question

Remark : Notice o product of three numbers is not equal EXERCISE 1.2 1. Express each number as a product of its prime factors: m 156 2. Find the LCM and HCFor the following pairs of integer product of the two numbers m 26 and 91 510 and 3. Find the LCM and HCF of the following integers by ap method 12. 15 and 2 17. 23 and 20 Given that HCF 306,657) = 9, find LCM (306.657)​

Answer 1

Answer:

(i) 26 and 91

Taking the LCM of 26,

26 = 2 × 13 × 1

Taking the LCM of 91,

91 = 7 × 13 × 1

Therefore, LCM of 26 and 91 together = 2 × 7 × 13 × 1 = 182

HCF of 26 and 91 = 13

Now, the product of 26 and 91 = 26 × 91 = 2366

And the product of LCM and HCF = 182 × 13 = 2366

Therefore, LCM × HCF = product of the 26 and 91.

(ii) 510 and 92

Taking the LCM of 510,

510 = 2 × 3 × 17 × 5 × 1

Taking the LCM of 92,

92 = 2 × 2 × 23 × 1

Therefore, LCM of 510 and 92 = 2 × 2 × 3 × 5 × 17 × 23 = 23460

HCF of 510 and 92 = 2

Now, the product of 510 and 92 = 510 × 92 = 46920

And the product of LCM and HCF = 23460 × 2 = 46920

Therefore, LCM × HCF = product of the 510 and 92.

(iii) 336 and 54

Taking the LCM of 336,

336 = 2 × 2 × 2 × 2 × 7 × 3 × 1

Taking the LCM of 54,

54 = 2 × 3 × 3 × 3 × 1

Therefore, LCM of 336 and 54 = 3024

HCF of 336 and 54 = 2×3 = 6

Now, the product of 336 and 54 = 336 × 54 = 18144

And the product of LCM and HCF = 3024 × 6 = 18144

Therefore, LCM × HCF = product of the 336 and 54

Answer 2

LCM × HCF = Product of the two numbers.

To find the LCM and HCF of the given pairs of the integers, first, find the prime factors of the given pairs of numbers.

Then, find the product of the smallest power of each common factor in the numbers. This will be the LCM.

Then, find the product of the greatest power of each prime factor in the number. This would be the HCF.

Now, we have to verify LCM × HCF = product of the two numbers.

(i) 26 and 91

Prime factors of 26 = 2 × 13

Prime factors of 91 = 7 × 13

HCF of 26 and 91 = 13

LCM of 26 and 91 = 2 × 7 × 13

= 14 × 13

= 182

Product of these two numbers = 26 × 91

= 2366

LCM × HCF = 182 × 13

= 2366

Thus, the product of two numbers = LCM × HCF

(ii) 510 and 92

Prime factors of 510 = 2 × 3 × 5 × 17

Prime factors of 92 = 2 × 2 × 23

HCF of the two numbers = 2

LCM of the two numbers = 2 × 2 × 3 × 5 × 17 × 23

= 23460

Product of these two numbers = 510 × 92

= 46920

LCM x HCF = 2 × 23460

= 46920

Thus, the product of two numbers = LCM × HCF

(iii) 336 and 54

Prime factors of 336 = 2 × 2 × 2 × 2 × 3 × 7

Prime factors of 54 = 2 × 3 × 3 × 3

HCF of the two numbers = 6

LCM of the two numbers = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 7

= 24 × 33 × 7

= 3024

Product of these two numbers = 336 × 54

= 18144

LCM x HCF = 3024 × 6

= 18144

Thus, the product of two numbers = LCM × HCF