Question

Prove that √5 is irrational Find the LCM and HCF prime factorization method. of the following integers by applying the 12,15 and 21​

Answer 1

Answer:

1) 12,15 and 21

we have 12=3×4

15=3×5

21=3×7

here,3 is the common factor.

so,the HCF of 12,15 and 21 is 3.

Now LCM = 3×4×5×7 =420

2) let's assume that √5 is rational.

Suppose a and b have a common factor other than 1 , than we can divide by the common factor, and assume that a and b are co-prime .

a and b (not = 0) such that

√5=a/b

so,b√5=a

squaring on both sides

5b^2 =a^2

Therefore a^2 is divisible by 5,and it follows that a is also divisible by 5.

So, a=5c,where c is some integer substituting for a, 5b^2= 25c^2

i,e- b^2=5c^2

This means that b^2 is divisible by 5 ,and so b is also divisible by 5.

Therefore a and b have at least 5 as a common factor .

But this contradicts the fact that a and b are co-prime.

because our contradiction is wrong.

so ,we conclude that √5 is irrational.

Answer 2

:

Mark Me As Brainliest And Thank Me If The Answer Is Useful.

-- :

Using prime factorisation method:

(i) 12, 15 and 21

Factor of 12=2×2×3

Factor of 15=3×5

Factor of 21=3×7

HCF (12,15,21)=3

LCM (12,15,21)=2×2×3×5×7=420

(ii) 17, 23 and 29

Factor of 17=1×17

Factor of 23=1×23

Factor of 29=1×29

HCF (17,23,29)=1

LCM (17,23,29)=1×17×23×29=11,339

(iii) 8, 9 and 25

Factor of 8=2×2×2×1

Factor of 9=3×3×1

Factor of 25=5×5×1

HCF (8,9,25)=1

LCM (8,9,25)=2×2×2×3×3×5×5=1,800