CBSE ADDA

Gist of lesson REAL NUMBER CLASS 10th • Euclid’s Division Lemma : Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. • Euclid’s Division Algorithm to obtain the HCF of two positive integers, say c and d, c > d. Step 1  : Apply Euclid’s division lemma to c and d, to 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 ¹ 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. • Fundamental Theorem of Arithmetic : Every composite number can be expressed as a product of primes, and this expression (factorisation) is unique, apart from the order in which the prime factors occur. • Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer. • √2 , √ 3 , √ 5 are irrational numbers. • Th

1. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer. 2. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons. 3. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer. 4. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer. 5. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer. 6. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer. 7. Explain why 3 × 5 × 7 + 7 is a composite number. 8. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons. 9. Without actually performing the long divisio

X Real Numbers : Topics: 1. Euclid's Division Lemma/Algorithm 2. Fundamental Theorem of Arithmetic 3. Irrational Numbers 4. Decimal expression of Rational Number Q.1. Based on Euclid’s algorithm: a = bq + r where  0 ≤ r Ð b Solved example:Using Euclid’s algorithm: Find the HCF of 825 and 175. Explanation: Step 1. Since 825>175. Divide 825 by 175. We get, quotient = 4 and remainder = 125. This can be written as 825 = 175 x 4 + 125 Step II. Now divide 175 by the remainder 125. We get quotient = 1 and remainder = 50. So we write 175 = 125 x 1 + 50. Step III. Repeating the above step we now divide 125 by 50 and get quotient = 2 and remainder = 25. so 125 = 50 x 2 + 25 Step IV. Now divide 50 by 25 to get quotient = 2 and remainder 0. Since remainder has become zero we stop here. Since divisor at this stage is 25, so the HCF of 825 and 175 is 25. Solution: This is how a student should write answer in his answer sheet: Since

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