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We have studied the following points:

1. Euclid’s Division Lemma : Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r where 0 ≤ r ≤ b.

2. Euclid’s Division Algorithm: According to this, which is based on Euclid’s division lemma, the HCF of any two positive integers a and b with a > b is obtained as follows:

Step 1 Apply the division lemma to find q and r where a = bq + r, O ≤ r < b.

Step 2 If r = 0, the HCF is b . If r ≠ 0 apply Euclid Lemma to b and r

Step 3 Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also HCF (a, b) = HCF (b, r)

3. The Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes and this factorisation is unique, apart from the order in which the prime factors occur.

4. To test whether a given rational number is a terminating or repeating decimal :
Let x be a rational number whose simplest form is , where p and q are integers and q ≠ 0 . Then

(i) x is a terminating decimal only when q is of the form 2n x 5m for some non-negative integer as m and n.

(ii) x is a non-terminating repeating decimal , if q 2n x 5m .

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