Euclid's Division Lemma Archives – Mathemafia

Class 10 | Chapter 1 | Real Numbers | Exercise 1.2

Sunil Syal — Mon, 19 Apr 2021 10:06:13 +0000

Question 1: Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Question 2: Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

Question 3: Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

Question 4: Given that HCF (306, 657) = 9, find LCM (306, 657).

Question 5: Check whether 6ⁿ can end with the digit 0 for any natural number n.

Question 6: Explain why 7×11×13 +13 and 7×6×5×4×3×2×1+5 are composite numbers.

Question 7: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

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Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Exercise 1.2 appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Exercise 1.1

Sunil Syal — Wed, 17 Mar 2021 03:51:12 +0000

Question 1: Use Euclid’s division algorithm to find the HCF of :
(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Question 2: Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.

Question 3: An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Question 4: Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Question 5: Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.

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🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Exercise 1.1 appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Examples

Sunil Syal — Fri, 05 Mar 2021 05:18:58 +0000

Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576.

Example 2: Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

Example 3: Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Example 4: A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?

Example 5: Consider the numbers 4ⁿ, where n is a natural number. Check whether there is any value of n for which 4ⁿ ends with the digit zero.

Example 6: Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Example 7: Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

Example 8: Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Example 9: Prove that √ 3 is irrational.

Example 10: Show that 5 – √ 3 is irrational.

Example 11: Show that 3√2 is irrational.

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– Saiba’s Parents (USA)

Amazing Math Coach

– Rishi’s Parents (USA)

Unique, Patient and Good Mentor

Also, your videos on Mathemafia youtube channel are very informative.

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

Real Numbers
- Exercise 1.1
- Exercise 1.2
Polynomials
- Exercise 2.1
- Exercise 2.2
Pair of Linear Equation in Two Variables

Quadratic Equations
Arithmetic Progressions
- Exercise 5.1
- Exercise 5.2
Triangles
- Exercise 6.1
- Exercise 6.2
- Exercise 6.3

Coordinate Geometry
- Exercise 7.1
- Exercise 7.2
Introduction to Trigonometry
Some Applications of Trigonometry
- Exercise 9.1
Circles
- Exercise 10.1
- Exercise 10.2

Areas Related to Circles
- Exercise 11.1
Surface Areas and Volumes
- Exercise 12.1
- Exercise 12.2
Statistics
- Exercise 13.1
- Exercise 13.2
- Exercise 13.3
Probability
- Exercise 14.1

Super Cool Stuff

Calculators
- Quadratic Equation Calculator – Solve any Equation
Super Concepts

Theorems
Quizzes

The post Class 10 | Chapter 1 | Real Numbers | Examples appeared first on Mathemafia.

Class 10 | Chapter 1 | Real Numbers | Theorems & Concepts

Sunil Syal — Fri, 05 Mar 2021 04:57:32 +0000

Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Euclid’s Division Algorithm : This algorithm is used to find the h.c.f (Highest Common Factor) or g.c.d (Greatest Common Divisor)of two positive integers. It is based on Euclid’s Division Lemma that has been stated and explained above.

Step 1 : Apply Euclid’s division lemma, to a and b to find integers (more precisely whole numbers) q and r such that a = bq + r, 0 ≤ r < b.
Step 2 : If r=0, b is the g.c.d of a and b. If r≠0, apply the division lemma to b and r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required GCD (HCF).

Theorem 1.3: Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

Theorem 1.4: √2 is irrational.

Theorem – Good to know: If a non -ve integer is not a perfect square, then its square root is irrational.

Theorem 1.5: Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2ⁿ5^m, where n, m are non-negative integers.

Theorem 1.6: Let x = p/q be a rational number, such that the prime factorisation q of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Theorem 1.7: Let x = p/q , where p and q are coprimes, be a rational number, such that the prime factorisation of q is not of the form 2ⁿ5^m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).