**1.** Given positive integers and , there exist unique integers and satisfying

. This result is known as Euclid’s division lemma.

**2.** An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

**3.** A lemma is a proven statement used for proving another statement.

**4. **HCF of two positive integers and is the largest positive integer d that divides both and .

**5. **Euclid’s Division Algorithm : To obtain the HCF of two positive integers, say and with , we follow the steps below :

**Step 1.** Apply Euclid’s division lemma to find and where .

**Step 2.** If , then is the HCF of and . If , then apply Euclid’s division lemma to and .

**Step 3.** Continue this process till the remainder is zero. The divisor at this stage will be the required HCF.

**6.** 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. Or the prime factorisation of a natural number is unique, except for the order of its factors.

**7. **Any number which cannot be expressed in the form where and are integers and is called an irrational number.

**8. **Let be prime number. If divides , then divides , where is a positive integer.

**9.** The sum or difference of a rational and an irrational number is irrational.

**10.** The product and quotient of a non-zero rational number and an irrational number is irrational.

**11.** Let be a rational number whose decimal expansion terminates. Then can be expressed in the form where and are coprime and the prime factorisation of is of the form , where and are non negative integers.

**12. **Let be a rational number such that the prime factorisation of is of the form , where and are non negative integers. Then, has a decimal expansion which terminates.

**13.** If is a rational number, such that the prime factorisation of is of the form , where and are whole numbers. If , then the decimal expansion of will terminate after places of decimal. If , then the decimal expansion of will terminate after places of decimal. If , then the decimal expansion of will terminate after places of decimal.

**14.** Let be a rational number, such that the prime factorisation of is not of the form , where and are non negative integers. Then has a decimal expansion which is non terminating repeating (recurring).

**15. **The decimal expansion of every rational number is either terminating or non-terminating repeating.

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