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# 1. Use Euclid’s division algorithm to find the HCF of 135 and 225.

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NCERT Solutions for Class 10 Chapter 1 Excercise 1.1

Question 1(!)

Difference Between Algorithm And Lemma.
Algorithm
An algorithm is a series of well-defined steps which gives a procedure for solving a type of problem.

Lemma
A lemma is a proven statement used for proving another statement.

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1. Step 1:  Since 225 > 135, apply Euclid’s division lemma, to a =225 and b=135 to find q and r   such that 225 = 135q+r, 0  r
On dividing 225 by 135 we get quotient as 1 and remainder as 90
i.e 225 = 135 x 1 + 90

Step 2: Remainder r which is 90  0, we apply Euclid’s division lemma to b =135 and r = 90 to find whole numbers q and r such that
135 = 90 x q + r,  0  r<90
On dividing 135 by 90 we get quotient as 1 and remainder as 45
i.e 135 = 90 x 1 + 45

Step 3:  Again remainder r = 45  0 so we apply Euclid’s division lemma to b =90 and r = 45 to find q and r   such that
90 = 90 x q + r,    0  r<45
On dividing 90 by 45 we get quotient as 2 and remainder as 0
i.e 90 = 2 x 45 + 0

Step 4: Since the remainder is zero, the divisor at this stage will be HCF of (135, 225).

Since the divisor at this stage is 45, therefore, the HCF of 135 and 225 is 45.

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