# Ex.1.1 Q1 Real Numbers Solution - NCERT Maths Class 10

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## Question

Use Euclid’s division algorithm to find the HCF of:

(i) $$135$$ and $$225$$

(ii) $$196$$ and $$38220$$

(iii) $$867$$ and $$255$$

Video Solution
Real Numbers
Ex 1.1 | Question 1

## Text Solution

What is known?

Two different numbers

What is unknown?

HCF of the given numbers.

Reasoning:

You have to find the HCF of given integers by using Euclid’s Division Lemma. It is a technique to compute the highest common factor of two given positive integer. Recall, that the HCF of two positive integers $$a$$ and $$b$$ is the largest positive integer that divides both $$a$$ and $$b.$$

To obtain the HCF of two positive integers say $$a$$ and $$b$$ with $$a > b,$$ follow the below steps-

Step- I. Apply Euclid’s division lemma to $$a$$ and $$b.$$ So, we find whole numbers $$q$$ and $$r$$ such that

$a = bq + r,\; 0 \le r < b$

Step - II.  If $$r=0,$$ $$b$$ is the HCF of $$a$$ and $$b.$$ If $$r \ne 0,$$ apply the division lemma to $$b$$ and $$r.$$

Step - III. Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

Steps:

(i) $$135$$ and $$225$$

In this case \begin{align} 225 > 135. \end{align} We apply Euclid’s division lemma to $$135$$ and $$225$$ and get

$225=(135 \times 1)+90$

Since, the remainder $$r \ne 0,$$ we apply the division lemma to $$135$$ and $$90$$ to get

$135=(90 \times 1)+45$

Now, we consider $$90$$ as the divisor and $$45$$ as the remainder and apply the division lemma, to get

$90 = (45\times 2 )+ 0$

Since, the remainder is zero and the divisor is $$45,$$ therefore, the H.C.F of $$135$$ and $$225$$ is $$45.$$

(ii)  $$196$$ and $$38220$$

$$38220$$ is greater than $$196,$$ we apply Euclid’s division lemma to $$38220$$ and $$196,$$ to get

$38220 = {{ }}(196{{ }} \times {{ }}195) + {{ }}0$

Since, the remainder is zero and the divisor in this step is $$195,$$ therefore, the H.C.F of $$38220$$ and $$196$$ is $$196.$$

(iii) $$867$$ and $$255$$

$$867$$ is greater than $$225$$ and on applying Euclid’s division lemma to $$867$$and $$225,$$ to get

$867 = {{ }}(255{{ }} \times {{ }}3) + {{ }}102$

Since, the remainder $$r{{ }} \ne {{ }}0,$$  we apply the division lemma to $$225$$ and $$102$$ and get

$255{ }=(102{ }\times { }2)+{ }51$

Again, remainder is not zero, we apply Euclid’s division lemma $$102$$ and $$51$$ which gives

$102=(51 \times \ 2)+0$

Since, the remainder is zero and the divisor is $$51,$$ therefore, the H.C.F of $$867$$ and $$255$$ is $$51.$$

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