All rounder knowledge

Exercise : 1.1

Question 1. Use Euclid's division algorithm to find HCF of

(i) 135 and 225

(ii) 196 and 38220

(iii) 867 and 255

Solution 1.

(i) 135 and 225

Since 225>135, we can apply Euclid's division lemma to 225 and 135

225 = 135×1 + 90

Since the remainder 90≠0 , we can apply Euclid's division lemma to 135 and 90

135 = 90×1 + 45

Again we can see that the remainder 45≠0 we will again apply Euclid's division lemma to 90 and 45

90 = 45×2 + 0

We get remainder = 0

So in last step divisor is 45 so the HCF of 135 and 225 is 45.

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Question 1.

Prove that √5 is irrational.

Solution:

Let √5 = p/q be a rational no. Where p and q are co primes and q≠0

Then √5q = p

Squaring both side we get :

=> 5q² = p² ............... (i)

Since 5 divides p² it will also divide p also

Let p = 5m ( where m is an integer )

Squaring both side we get :

=> P² = 25q²

=> 5q² = 25p² ....... ( From i )

=> q² = 5p²

Since 5 divides q² it will also divide q

So 5 is a common factor which contradict the fact that p and q are co primes.

Thus our assumption that  √5 is rational no. Is wrong

Hence √5 is an irrational no.

Hence proved.

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Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 Solution : (i) 140

Prime factors : 2×2×5×7

(ii) 156

Prime factors : 2×2×3×13

(iii) 3825

Prime factors : 3×3×5×5×17

(iv) 5005

Prime factors : 5×7×11×13

(v) 7429

Prime factors :  17×19×23

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