Very Short Answer Type Questions: Real Numbers

# Class 10 Maths Chapter 1 Question Answers - Real Numbers

Q1: Given that HCF (150, 100) = 50. Find LCM (150, 100).
Sol: LCM × HCF  = Product of the two numbers
∴  150 × 100 =  LCM × HCF
⇒ LCM × 50  =  150 × 100
⇒

Q2: Given that LCM (26, 91) = 182. Find their HCF.
Sol: ∵  HCF × LCM  =  Product of the two numbers
∴   HCF × 182  =  26 × 91
⇒

Q3: The LCM and HCF of the two numbers are 240 and 12 respectively. If one of the numbers is 60, then find the other number.
Sol: Let the required number be ‘x’.
∵  LCM × HCF   =  Product of the two numbers
∴ 60 × x  = 240 × 12

Q4: The decimal expansion of the rational number, will terminate after how many places of decimal?
Sol:

Thus,   will terminate after 4 places of decimal.

Q5: What is the exponent of 3 in the prime factorisation of 864.
Sol:

Making prime factors of 864. ,⇒ 864 = 3 × 3 × 3 × 2 × 2 × 2 × 2 × 2= 33 × 25
∴ Exponent of 3 in prime factorisation of 864 = 3.

Q6: State the fundamental theorem of arithmetic.
Sol: Every composite number can be expressed as the product of primes and this decomposition is unique apart from the order in which prime factors occur.

Q7:  Define an irrational number.
Sol: Those numbers which neither terminate in their decimal expansion nor can be expressed as recurring decimals are irrational numbers i.e., the numbers which cannot be expressed as p/q form (q ≠ 0), are called irrational numbers.

Q8: Write the condition for a rational number which can have a terminating decimal expansion.
Sol: A rational number x = p/q can have a terminating decimal expansion if the prime factorisation of q is of the form of 2n · 5m, where m and n are non-negative integers.

Q9: Write the condition for a rational number which has a non-terminating repeating decimal expansion.
Sol: A rational number x = p/q can have a non-terminating repeating decimal expansion if the prime factorisation of q is not of the form 2n · 5m, where n, m are non-negative integers.

Q10: Can two numbers have 24 as their HCF and 7290 as their LCM? Give reasons.
Sol: No, because HCF always divides LCM but here 24 does not divide 7290.

Q11: If 6n is a number such that n is a natural number. Check whether there is any value of n ∈ N for which 6n is divisible by 7.
Sol: ∵ 6 = 2 × 3
∴  6n =  (2 × 3)n = 2n × 3n
i.e., the prime factorisation of 6n does not contain the prime number 7 thus the number 6n is not divisible by 7.

Q12: Write 98 as the product of its prime factors.
Sol:  ∵

∴ 98 = 2 × 7 × 7 ⇒  98 = 2 × 72

Q13: Without actually performing the long division, state whether  will have a terminating or non-terminating repeating decimal expansion.
Sol: Let  =
∵ Prime factors of q are not of the for 2n · 5m.
will have a non-terminating repeating decimal expansion.

Q14: Without actually performing the long division, state whether 17/3125 will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
Sol: ∵ The denominator of 17/3125 is given by
3125 =    5 × 5 × 5 × 5 × 5
=  1 × 55
= 2× 55    |∵ 20 = 1
∴

i.e., 17/3125 is a terminating decimal.

Q15: Express 156 as a product of its prime factors.
Sol:  ∵  156 = 2 × 78
= 2 × 2 × 39
=  2 × 2 × 3 × 13
∴ 156 = 22 × 3 × 13

Q16: If the product of two numbers is 20736 and their LCM is 384, find their HCF.
Sol: ∵  LCM × HCF = Product of two numbers
∴  384 × HCF =  20736
⇒ HCF  = 20736 /384   = 54.

Q17: Find the LCM and HCF of 120 and 144 by the Fundamental Theorem of Arithmetic.
Sol: We have 120 =  2 × 2 × 2 × 3 × 5 = 2× 3 × 5
144 =  2 × 2 × 2 × 2 × 3 × 3 = 24 × 32
∴ LCM  =  24 × 32 × 5 = 720
HCF  =  23 × 3 = 24

Q18: Find the HCF × LCM for the numbers 100 and 190.
Solution:  HCF × LCM  =  1st Number × 2nd Number
=  100 × 190 = 19000.

Q19: Find the (HCF × LCM) for the numbers 105 and 120.
Solution:  HCF × LCM =  1st number × 2nd number
= 105 × 120 = 12600.

Q20: Write a rational number between √2 and √3.
Sol:√2  =  1.41 ..... and
√3  =  1.73 .....
∴ one rational number between 1.41 .....and 1.73 ..... is 1.5
i.e., one rational number between √2 and √3 is 1.5.

The document Class 10 Maths Chapter 1 Question Answers - Real Numbers is a part of the Class 10 Course Mathematics (Maths) Class 10.
All you need of Class 10 at this link: Class 10

## Mathematics (Maths) Class 10

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## FAQs on Class 10 Maths Chapter 1 Question Answers - Real Numbers

 1. What are real numbers?
Ans. Real numbers are a set of numbers that includes both rational and irrational numbers. They can be expressed as decimal numbers or fractions, and they extend infinitely in both positive and negative directions on the number line.
 2. How can you determine if a number is a real number?
Ans. Any number that can be expressed as a decimal or a fraction is a real number. Additionally, numbers like √2 or π, which cannot be expressed as fractions, are also considered real numbers. In general, if a number exists on the number line, it is a real number.
 3. What is the difference between rational and irrational numbers in the context of real numbers?
Ans. Rational numbers are numbers that can be expressed as fractions, where the numerator and denominator are both integers. On the other hand, irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal representations.
 4. How do real numbers relate to everyday life?
Ans. Real numbers are used in various aspects of everyday life, such as measuring quantities, calculating distances, and representing values in financial transactions. They are essential in fields like physics, engineering, economics, and many other disciplines that involve mathematical calculations.
 5. Can you give examples of irrational numbers?
Ans. Yes, some examples of irrational numbers include √2, π (pi), and e (Euler's number). These numbers cannot be expressed as fractions and have decimal representations that go on infinitely without repeating.

## Mathematics (Maths) Class 10

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