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Euclid's Division Lemma Euclid’s division lemma, states that for any two positive integers „a‟ and „b‟ we can find two whole numbers „q‟ and „r‟ such that Euclid’s division lemma can be used to: Find the highest common factor of any two positive integers and to show the common properties of numbers. Finding H.C.F using Euclid’s division lemma: Suppose, we have two positive integers „a‟ and „b‟ such that „a‟ is greater than „b‟. Apply Euclid’s division lemma to the given integers „a‟ and „b‟ to find two whole numbers „q‟ and „r‟ such that, „a‟ is equal to „b‟ multiplied by „q‟ plus „r‟. Check the value of „r‟. If „r‟ is equal to zero then „b‟ is the HCF of the given numbers. If „r‟ is not equal to zero, apply Euclid’s division lemma to the new divisor „b‟ and remainder „r‟. Continue this process till the remainder „r‟ becomes zero. The value of the divisor „b‟ in that case is the HCF of the two given numbers. Euclid’s division algorithm can also be used to find some common properties of numbers Fundamental Theorem of Arithmetic Fundamental Theorem of Arithmetic states that: Every composite number can be expressed or factorised as a product of prime numbers and this factorisation is unique except in the order of the prime factors. We can write the prime factorisation of a number in the form of powers of its prime factors. By expressing any two numbers as their prime factors, their highest common factor (HCF) and lowest common multiple (LCM) can be easily calculated. The HCF of two numbers is equal to the product of the terms containing the least powers of common prime factors of the two numbers. The LCM of two numbers is equal to the product of the terms containing the greatest powers of all prime factors of the two numbers. Note that the product of the given numbers is equal to the product of their HCF and LCM. This result is true for all positive integers and is often used to find the HCF of two given numbers if their LCM is given and vice versa Revisiting Rational and Irrational Numbers A number is called a rational number if it can be written in the form a/b where a and b are integers and b ≠ 0. A number is called an irrational number if it cannot be written in the form a/b, where a and b are integers and b ≠ 0. The sum, difference, product or quotient of a rational and an irrational number is also an irrational number. Rational numbers are of two types depending on whether their decimal form is terminating or recurring. Theorem: If p/q is a rational number, such that the prime factorisation of q is of the form 2a5b, where a and b are positive integers, then the decimal expansion of the rational number p/q terminates. Theorem: If a rational number is a terminating decimal, it can be written in the form p/q, where p and q are co prime and the prime factorisation of q is of the form 2a5b, where a and b are positive integers. Theorem: If p/q is a rational number such that the prime factorisation of q is not of the form 2a5b where a and b are positive integers, then the decimal expansion of the rational number p/q does not terminate and is recurring