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Introduction

Real Numbers

A Real number is any element of the set R, which is the union of the set of Natural numbers N, Integer numbers Z, Rational numbers Q and the set of Irrational numbers.

The largest common factor of two or more numbers is called the highest common factor (HCF).

A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24, .... The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.

The greatest common divisor (gcd) of two or more integers, when at least one of them is not zero, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.

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

Euclid's theorem or Euclid's algorithm. In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers,
namely: If a prime divides the product of two numbers, it must divide at least one of those numbers.

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

**Step 1 :** Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.

**Step 2 :** If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.

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

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.

Prime Factorization is finding which prime numbers multiply together to make the original number.

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