Two core concepts tested in this chapter are LCM and HCF. Watch these three videos to understand the two concepts and **when to use LCM and when to use HCF**?

Chapter 1 of CBSE NCERT Class 10 Math covers Real Numbers. Concepts covered in chapter 1 include Euclid's Division Lemma, Fundamental Theorem of Arithmetic and Prime Factorization, Highest Common Factor(HCF), Lowest Common Multiple(LCM), Product of HCF and LCM, Prime Factorisation, Irrational Numbers, and Rational Numbers. The extra questions given below include questions akin to HOTS (Higher Order Thinking Skills) questions and exemplar questions of NCERT.

Here is a quick recap of the key concepts that are covered in this chapter in the CBSE NCERT Class 10 Math text book. Already, know these concepts? Jump to __extra questions__ directly.

If we have two positive integers n and d, then there would be whole numbers q and r that satisfy the equation: **n= qd + r**, where 0 ≤ r < d. n is the dividend. q is the quotient. d is the divisor.

The Fundamental Theorem of Arithmetic states that, *"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."*

Highest Common Factor (HCF) of two or more numbers is the largest number that divides all of those numbers without leaving a remainder.

Highest Common Factor (HCF) is the product of the lowest power of common prime factors in the numbers.

Euclid's Division Algorithm can be used to compute the Highest Common Factor (HCF) of the two or more positive integers. Let the given positive integers be a and b such that a > b

**Step 1:** Apply Euclid's Division Lemma to find q and r where a = bq + r, 0 ≤ r < b.

**Step 2:** If r = 0, then the HCF is b. If r ≠ 0, apply Euclid's Lemma to b and r.

**Step 3:** Continue the process till the remainder is zero. The divisor at the step in which the remainder is zero is the HCF(a, b). Also HCF(a, b) = HCF(b, r).

Lowest Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of those numbers.

Lowest Common Multiple (LCM) is the product of the greatest power of all the prime factors found in the numbers.

A number ‘s’ is irrational if it cannot be written in the form \\frac{\text{p}}{\text{q}}), where p and q are integers and q ≠ 0.

The decimal expansion of an irrational number is non-terminating and non-recurring.

Question 1

Karan has 180 blue marbles and 150 red marbles. He wants to pack them into packets containing equal number of marbles of the same colour. What is the maximum number of marbles that each packet can hold?

Question 2

Question 3

Question 4

Question 5

What is the largest number that divides 170, 220, and 420 leaving remainder 8, 4 and 15 respectively?

Question 6

Find the LCM and HCF of the following :

(i) 2^{5}× 5^{4}× 7^{2}× 13^{6}and 2^{3}× 5^{6}× 7 × 17^{3}

(ii) a^{5}× b^{2}× c^{2}× d^{5}and a^{7}× b^{3}× e × f^{3}where a, b, c, d, e, and f are prime.

Question 7

The Muscle Gym has bought 63 treadmills and 108 elliptical machines. The gym divides them into several identical sets of treadmills and elliptical machines for its branches located throughout the city, with no exercise equipment left over. What is the greatest number of branches the gym can have in the city?

Question 8

Katya has 49 paintings and 35 medals. She wants to display them in groups throughout her house, each with the same combination of paintings and medals, with none left over. What is the greatest number of groups Katya can display?

Question 9

Anish goes fishing every 5th day and Balaji goes fishing every 7th day. If Anish and Balaji both went fishing today, how many days until they will go fishing on the same day again?

Question 10

Tamanna is arranging black marbles in groups of 13 and purple marbles in groups of 25. If she has the same number of black and purple marbles, what is the smallest number of marbles of each colour that she could have?

Class 10 Maths

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CBSE Class 10 Maths - 2021

- Real Numbers Revision Class
- Polynomials Revision Videos
- Linear Equations Revision Class
- Quadratic Equations Revision Class
- Arithmetic Progressions Revision Video
- Triangles Revision
- Coordinate Geometry Revision
- Trigonometry Revision Class
- Appl of Trigonometry
- Circles Revision Class
- Areas Related to Circles Revision Videos

1. HCF Word Problem | CBSE Class 10 Math Extra Practice Question **Video Solution **▶

2. Highest Common Factor of 2 Numbers | CBSE 10th Math Extra Practice Question **Video Solution** ▶

3. HCF of 3 numbers | CBSE Class 10 Math Extra Practice Question **Video Solution** ▶

4. Common divisor - different remainders - 2 numbers | CBSE 10th Math Extra Practice Question **Video Solution** ▶

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