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 < bEuclid’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 = p1p2 ..... pn , where p1p2 ..... pn are primes and written in ascending order, i.e., p1≤ p2≤ ..... ≤ pn. 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 = 23× 32× 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.