Extra Questions for Class 10 Maths Real Numbers

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?

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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.

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What is Euclid’s Division Lemma

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.

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Fundamental Theorem of Arithmetic

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."

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What is HCF? How to find HCF?

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.

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How to use Euclid’s Division Lemma to compute HCF?

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).

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What is LCM? How to find LCM?

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.

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What are irrational 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.

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Extra Questions for Class 10 Maths - Real Numbers

1. 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?

   Solution Video Solution

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2. Question 2

   Find the largest number that will divide 382 and 710 and leaves a remainder 13 in each case.

   Solution Video Solution

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3. Question 3

   What is the largest number that divides 437, 732, and 1263 leaving remainder of 24 in each case?

   Solution Video Solution

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4. Question 4

   What is the largest number that divides 967 and 1767 leaving remainders of 71 and 103 respectively?

   Solution Video Solution

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5. Question 5

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

   Solution Video Solution

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6. Question 6

   Find the LCM and HCF of the following :
   (i) 2⁵ × 5⁴ × 7² × 13⁶ and 2³ × 5⁶ × 7 × 17³
   (ii) a⁵ × b² × c² × d⁵ and a⁷ × b³ × e × f³ where a, b, c, d, e, and f are prime.

   Solution Video Solution

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7. 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?

   Solution Video Solution

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8. 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?

   Solution Video Solution

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9. 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?

   Solution Video Solution

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10. 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?

    Solution Video Solution

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