#### Study Material and Revision Notes of Ch 1 Real Numbers Class 10th Maths

**Topics in the Chapter**

• Euclid’s Division Lemma

• Euclid’s Division Algorithm

• The Fundamental Theorem of Arithmetic

• Revisiting Irrational Numbers

• Revisiting Rational Numbers and Their Decimal Expansions

**Euclid’s Division Lemma**

→ For any two positive integers a and b, there exist unique integers q and r such that

*a = bq + r, 0 ≤ r < b*

Here, a, b, q and r are called dividend, divisor, quotient and remainder respectively.

→ Example: Dividing 34 by 6, we get quotient as 6 and remainder as 4.

So we can write it as, 34 = 6×5 + 4

Here, a = 34, b = 6, q = 5 and r = 4

∴ *a = bq + r, 0 ≤ r < b*

**Euclid’s Division Algorithm**

→ Euclid’s Division Algorithm is application of Euclid’s Division Lemma means it is technique to compute HCF of two natural numbers.

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

Step 1: Apply Euclid’s division lemma to c and d to obtain two whole numbers q and r.

*Thus,*

*c = dq + r, 0 ≤ r < d*

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

Step 3 : Continue the process till the remainder is zero. The last divisor when remainder becomes zero is the HCF of c and d

→ Example: Use Euclid’s division algorithm to find the HCF of 420 and 130.

Here, 420 > 130 so we can write,

Step 1: 420 = 130 × 3 + 30 (remainder is 30 so we will apply division lemma again on 130 and 30)

Step 2: 130 = 30 × 4 + 10 (remainder is 10 so we will apply division lemma again on 30 and 10)

Step 3: 30 = 10 × 3 + 0

Now, the remainder is 0 and the last divisor is 10. So, 10 is the HCF of 420 and 130.

**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.**

→ Let x be composite number which can be factorised as,

x = p

_{1}p

_{2}..... p

_{n}, where p

_{1}p

_{2}..... p

_{n}are primes and written in ascending order, i.e., p

_{1}≤ p

_{2}≤ ..... ≤ p

_{n}. If we combine the same primes, we will get powers of primes. This is called prime factorization.

For example: 32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 2

^{3}× 3

^{2}× 5 × 7 × 13

By expressing any two numbers as their prime factors, their highest common factor (HCF) and lowest common multiple (LCM) can be calculated.