# Summary

**We have studied the following points:**

1. Euclidâ€™s Division Lemma : Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r where 0 â‰¤ r â‰¤ b.

2. Euclidâ€™s Division Algorithm: According to this, which is based on Euclidâ€™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 â‰¤ r < b.

**Step 2** If r = 0, the HCF is b . If r â‰ 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 â‰ 0 . Then

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

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