# IMPORTANT TERMS, DEFINITIONS AND RESULTS

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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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