CBSE Class 10 Mathematics (2026–27)
Chapter 1: Real Numbers
20 Important Questions and Answers
Q1. What is Euclid’s Division Lemma? Explain with an example.
Answer:
Euclid’s Division Lemma is a mathematical statement used to divide two positive integers. It states that for any two positive integers a and b, there exist unique whole numbers q (quotient) and r (remainder) such that:
a = bq + r, where 0 ≤ r < b.
This lemma forms the basis of the Euclidean Algorithm for finding the HCF of two numbers. For example, if 29 is divided by 5, then:
29 = 5 × 5 + 4
Here, 29 is the dividend, 5 is the divisor, 5 is the quotient, and 4 is the remainder. Since the remainder is less than the divisor, the division satisfies Euclid’s Division Lemma.
Q2. Explain the Euclidean Algorithm for finding HCF.
Answer:
The Euclidean Algorithm is a method used to find the Highest Common Factor (HCF) of two positive integers using repeated division. According to Euclid’s Division Lemma, divide the larger number by the smaller number. Then divide the divisor by the remainder. Continue the process until the remainder becomes zero. The last non-zero remainder is the HCF.
For example, to find the HCF of 56 and 42:
56 = 42 × 1 + 14
42 = 14 × 3 + 0
Since the remainder becomes zero, the HCF is 14. This method is efficient and widely used in number theory. It helps simplify calculations involving large numbers.
Q3. Find the HCF of 135 and 225 using Euclid’s Division Algorithm.
Answer:
To find the HCF of 135 and 225:
225 = 135 × 1 + 90
135 = 90 × 1 + 45
90 = 45 × 2 + 0
Since the remainder becomes zero, the last non-zero remainder is 45.
Therefore, HCF (135, 225) = 45.
Euclid’s Division Algorithm provides a systematic and quick method to determine the HCF of large numbers. Instead of listing all factors, repeated division helps reach the answer efficiently. The HCF is useful in simplifying fractions, solving word problems, and finding common measures. Hence, the HCF of 135 and 225 is 45.
Q4. Define prime numbers and composite numbers with examples.
Answer:
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, and 11. The number 2 is the only even prime number.
A composite number is a natural number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, and 12.
For example, 7 is prime because its only factors are 1 and 7. On the other hand, 12 is composite because it has factors 1, 2, 3, 4, 6, and 12.
Prime numbers play an important role in factorization and number theory.
Q5. What is the Fundamental Theorem of Arithmetic?
Answer:
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers, and this factorization is unique except for the order of the factors.
For example:
60 = 2 × 2 × 3 × 5
or
60 = 2² × 3 × 5
No other set of prime factors can produce 60. The arrangement of factors may change, but the prime factors remain the same.
This theorem forms the basis of many mathematical concepts, including HCF, LCM, irrational numbers, and divisibility rules. It helps us understand the structure of numbers and is one of the most important results in elementary number theory.
Q6. Find the prime factorization of 540.
Answer:
To find the prime factorization of 540, divide it repeatedly by prime numbers:
540 = 2 × 270
= 2 × 2 × 135
= 2 × 2 × 3 × 45
= 2 × 2 × 3 × 3 × 15
= 2 × 2 × 3 × 3 × 3 × 5
Therefore,
540 = 2² × 3³ × 5
This representation shows the number as a product of prime factors only. Prime factorization is useful in finding HCF, LCM, and simplifying mathematical expressions. According to the Fundamental Theorem of Arithmetic, this factorization is unique except for the order of factors.
Q7. How can HCF be found using prime factorization?
Answer:
The HCF of two numbers can be found using prime factorization by following these steps:
- Express each number as a product of prime factors.
- Identify the common prime factors.
- Take the lowest power of each common prime factor.
- Multiply these factors.
For example:
36 = 2² × 3²
48 = 2⁴ × 3
Common factors are 2 and 3.
Lowest powers are 2² and 3¹.
HCF = 2² × 3 = 4 × 3 = 12
Thus, the HCF of 36 and 48 is 12. This method is particularly useful when dealing with larger numbers and understanding factor relationships.
Q8. How can LCM be found using prime factorization?
Answer:
To find the LCM using prime factorization:
- Express each number as a product of prime factors.
- Take all prime factors involved.
- Select the highest power of each prime factor.
- Multiply them together.
For example:
24 = 2³ × 3
36 = 2² × 3²
LCM = 2³ × 3²
= 8 × 9
= 72
Thus, the LCM of 24 and 36 is 72.
The LCM is the smallest positive number divisible by both given numbers. Prime factorization provides a systematic method for calculating LCM accurately and efficiently.
Q9. Verify that HCF × LCM = Product of two numbers for 18 and 24.
Answer:
For the numbers 18 and 24:
Prime factorization:
18 = 2 × 3²
24 = 2³ × 3
HCF = 2 × 3 = 6
LCM = 2³ × 3² = 72
Now,
HCF × LCM = 6 × 72 = 432
Product of the numbers:
18 × 24 = 432
Since both values are equal,
HCF × LCM = Product of the two numbers
This verifies the relationship. This property is applicable to any pair of positive integers and is useful for checking the correctness of HCF and LCM calculations.
Q10. What are irrational numbers? Give examples.
Answer:
Irrational numbers are numbers that cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. Their decimal expansions are non-terminating and non-recurring.
Examples include:
- √2
- √3
- √5
- π
For example, √2 = 1.41421356… continues endlessly without repeating any pattern.
Unlike rational numbers, irrational numbers cannot be written as fractions. They are important in geometry, algebra, and real-life measurements. Many square roots of non-perfect squares are irrational. Together with rational numbers, they form the set of real numbers.
Q11. Show that √5 is irrational.
Answer:
Assume √5 is rational.
Then √5 = p/q, where p and q are coprime integers.
Squaring both sides:
5 = p²/q²
⇒ p² = 5q²
This means p² is divisible by 5, so p is also divisible by 5.
Let p = 5k.
Substituting:
25k² = 5q²
⇒ q² = 5k²
Therefore q is also divisible by 5.
This means p and q have a common factor 5, which contradicts the assumption that they are coprime.
Hence, the assumption is false.
Therefore, √5 is an irrational number.
Q12. What are terminating decimals?
Answer:
A terminating decimal is a decimal number that has a finite number of digits after the decimal point. Such decimals end after a certain number of places.
Examples:
- 0.5
- 0.75
- 2.125
These numbers can be expressed as fractions whose denominators have only factors 2 and/or 5 after simplification.
For example:
3/8 = 0.375
Since 8 = 2³, the decimal terminates.
Terminating decimals are rational numbers and can always be written as fractions. Understanding them helps classify numbers and determine whether decimal expansions end or continue indefinitely.
Q13. What are non-terminating recurring decimals?
Answer:
Non-terminating recurring decimals are decimal numbers that continue forever but have a repeating pattern of digits.
Examples:
- 0.3333… = 1/3
- 0.272727… = 27/99
The repeating part is called a recurring block. Although these decimals never end, they are rational numbers because they can be expressed as fractions.
For instance:
0.7777… = 7/9
Recurring decimals differ from irrational numbers because irrational numbers neither terminate nor repeat. Identifying recurring decimals helps classify numbers correctly and convert them into fractions when required.
Q14. State the condition for a rational number to have a terminating decimal expansion.
Answer:
A rational number p/q, where q ≠ 0 and p and q are coprime, has a terminating decimal expansion if the prime factorization of q contains only the prime factors 2 and/or 5.
Examples:
1/8 = 0.125
1/20 = 0.05
Since:
8 = 2³
20 = 2² × 5
both decimals terminate.
However, if the denominator contains any prime factor other than 2 or 5, the decimal expansion will be non-terminating recurring.
This condition helps determine the nature of decimal expansions without performing actual division.
Q15. Determine whether 13/125 has a terminating decimal expansion.
Answer:
Consider the rational number:
13/125
Prime factorization of the denominator:
125 = 5³
The denominator contains only the prime factor 5.
According to the condition for terminating decimals, if the denominator after simplification contains only 2 and/or 5 as prime factors, the decimal expansion terminates.
Now dividing:
13 ÷ 125 = 0.104
Since the decimal ends after three places, it is a terminating decimal.
Therefore, 13/125 has a terminating decimal expansion. This conclusion can also be reached directly from the denominator’s prime factorization.
Q16. Determine whether 7/24 has a terminating decimal expansion.
Answer:
Consider the rational number:
7/24
Prime factorization of the denominator:
24 = 2³ × 3
The denominator contains the prime factor 3 in addition to 2.
According to the theorem on decimal expansions, a rational number has a terminating decimal only when the denominator contains only 2 and/or 5 as prime factors.
Since 24 contains 3, the decimal expansion of 7/24 will not terminate.
In fact:
7/24 = 0.291666…
The digit 6 repeats endlessly.
Therefore, 7/24 has a non-terminating recurring decimal expansion.
Q17. Explain the relationship between rational and irrational numbers.
Answer:
Real numbers are divided into two categories: rational and irrational numbers.
Rational numbers can be expressed as p/q, where p and q are integers and q ≠ 0. Examples include 1/2, 3, and 0.75.
Irrational numbers cannot be expressed as fractions. Their decimal expansions are non-terminating and non-recurring. Examples include √2 and π.
Together, rational and irrational numbers form the set of real numbers. Every point on the number line represents a real number. Understanding the distinction between these two types helps in solving algebraic and numerical problems effectively.
Q18. Why is 2 the only even prime number?
Answer:
A prime number has exactly two factors: 1 and itself. The number 2 satisfies this condition because its factors are only 1 and 2.
Every other even number is divisible by 2 in addition to 1 and itself. Therefore, it has at least three factors and cannot be prime.
For example:
4 has factors 1, 2, and 4.
6 has factors 1, 2, 3, and 6.
Hence, these numbers are composite.
Since 2 is the only even number with exactly two factors, it is the only even prime number. This makes it unique among all prime numbers.
Q19. What is a composite number? Give examples.
Answer:
A composite number is a natural number greater than 1 that has more than two factors. In other words, it is divisible by numbers other than 1 and itself.
Examples:
- 4 = 1, 2, 4
- 6 = 1, 2, 3, 6
- 9 = 1, 3, 9
Since these numbers have more than two factors, they are composite.
Every composite number can be expressed as a product of prime factors. Composite numbers are important in factorization, HCF, and LCM calculations. Understanding them helps distinguish them from prime numbers.
Q20. Why are real numbers important in mathematics?
Answer:
Real numbers include all rational and irrational numbers. They represent every possible point on the number line and are widely used in mathematics and daily life.
Examples of real numbers include:
- Integers: -3, 0, 5
- Fractions: 2/3
- Decimals: 1.25
- Irrational numbers: √2, π
Real numbers are used in measurement, geometry, algebra, statistics, and science. They help represent quantities such as distance, weight, temperature, and time. Understanding real numbers is essential because they form the foundation for higher mathematical concepts and practical problem-solving in everyday situations.