NCERT Class 10 Maths Solutions: Real Numbers

Question:

Prime factorization is the process of breaking down a composite number into its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To find the prime factors of a number, we can use trial division, starting with the smallest prime numbers and checking if they divide the given number evenly.


We need to express the number 7429 as a product of its prime factors. We will use trial division to find these factors.

Step 1: Start by testing the smallest prime numbers.
Check divisibility by 2: 7429 is an odd number, so it is not divisible by 2.
Check divisibility by 3: Sum of the digits of 7429 is 7 + 4 + 2 + 9 = 22. Since 22 is not divisible by 3, 7429 is not divisible by 3.
Check divisibility by 5: 7429 does not end in 0 or 5, so it is not divisible by 5.
Check divisibility by 7: 7429 ÷ 7 = 1061 with a remainder. So, it is not divisible by 7.
Check divisibility by 11: Alternating sum of digits is (9 + 4) – (2 + 7) = 13 – 9 = 4. Since 4 is not divisible by 11, 7429 is not divisible by 11.
Check divisibility by 13: 7429 ÷ 13 = 571 with a remainder. So, it is not divisible by 13.
Check divisibility by 17: 7429 ÷ 17 = 437. So, 17 is a prime factor.

Step 2: Now we need to find the prime factors of 437.
We continue testing prime numbers starting from 17 (since we already checked smaller primes).
Check divisibility by 17: 437 ÷ 17 = 25 with a remainder. So, it is not divisible by 17.
Check divisibility by 19: 437 ÷ 19 = 23. So, 19 is a prime factor.

Step 3: Now we need to check if 23 is a prime number.
23 is only divisible by 1 and 23, so it is a prime number.

Step 4: Combine the prime factors.
We found that 7429 = 17 × 437 and 437 = 19 × 23.
Therefore, the prime factorization of 7429 is 17 × 19 × 23.

The number 7429 can be expressed as the product of its prime factors: 17 × 19 × 23.

Question:

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1 that have only two divisors: 1 and themselves. To find the prime factors of a number, we repeatedly divide the number by the smallest prime number that divides it evenly until the quotient is a prime number.


To express 3825 as a product of its prime factors, we will follow these steps:

1. Start with the smallest prime number, which is 2. Check if 3825 is divisible by 2. Since 3825 is an odd number (its last digit is 5), it is not divisible by 2.

2. Move to the next smallest prime number, which is 3. To check if 3825 is divisible by 3, sum its digits: 3 + 8 + 2 + 5 = 18. Since 18 is divisible by 3 (18 / 3 = 6), 3825 is also divisible by 3.
3825 ÷ 3 = 1275

3. Now, we need to find the prime factors of 1275. Check if 1275 is divisible by 3. Sum its digits: 1 + 2 + 7 + 5 = 15. Since 15 is divisible by 3 (15 / 3 = 5), 1275 is also divisible by 3.
1275 ÷ 3 = 425

4. Now, we need to find the prime factors of 425. Check if 425 is divisible by 3. Sum its digits: 4 + 2 + 5 = 11. Since 11 is not divisible by 3, 425 is not divisible by 3.

5. Move to the next smallest prime number, which is 5. Check if 425 is divisible by 5. Since the last digit of 425 is 5, it is divisible by 5.
425 ÷ 5 = 85

6. Now, we need to find the prime factors of 85. Check if 85 is divisible by 5. Since the last digit of 85 is 5, it is divisible by 5.
85 ÷ 5 = 17

7. Now, we need to find the prime factors of 17. 17 is a prime number because it is only divisible by 1 and itself.

Therefore, the prime factorization of 3825 is the product of all the prime numbers we divided by and the final prime quotient.
3825 = 3 × 3 × 5 × 5 × 17

We can also write this using exponents:
3825 = 3² × 5² × 17

Question:

The proof of irrationality for numbers of the form a + sqrt(b) or a * sqrt(b) (where a is rational and b is a non-perfect square positive integer) relies on the principle of contradiction. We assume the number is rational, and then use the properties of rational and irrational numbers to show that this assumption leads to a contradiction, thus proving the original number must be irrational. Key properties used are:
1. A rational number can be expressed as p/q, where p and q are integers and q is not zero.
2. The sum or difference of a rational number and an irrational number is irrational.
3. The product of a non-zero rational number and an irrational number is irrational.


We want to prove that $6+\sqrt{2}$ is irrational.
We will use the method of contradiction.
Assume that $6+\sqrt{2}$ is a rational number.
If $6+\sqrt{2}$ is rational, then it can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0$, and $p$ and $q$ are coprime (have no common factors other than 1).
So, $6+\sqrt{2} = \frac{p}{q}$.
Now, rearrange the equation to isolate $\sqrt{2}$:
$\sqrt{2} = \frac{p}{q} – 6$.
To combine the terms on the right side, we find a common denominator:
$\sqrt{2} = \frac{p}{q} – \frac{6q}{q}$.
$\sqrt{2} = \frac{p – 6q}{q}$.
Since $p$ and $q$ are integers, $p – 6q$ is also an integer.
Since $q$ is a non-zero integer, $\frac{p – 6q}{q}$ is a rational number.
This means that $\sqrt{2}$ is a rational number.
However, we know that $\sqrt{2}$ is an irrational number.
This is a contradiction.
Therefore, our initial assumption that $6+\sqrt{2}$ is rational must be false.
Hence, $6+\sqrt{2}$ is an irrational number.

Question:

Prime Factorisation Method: Breaking down a number into its prime factors.
LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more integers. To find LCM using prime factorization, take the highest power of each prime factor present in any of the numbers.
HCF (Highest Common Factor) or GCD (Greatest Common Divisor): The largest positive integer that divides two or more integers without leaving a remainder. To find HCF using prime factorization, take the lowest power of each common prime factor present in all the numbers.


To find the LCM and HCF of 8, 9, and 25 using the prime factorization method, we first find the prime factors of each number:

Prime factorization of 8:
8 = 2 × 2 × 2 = 2³

Prime factorization of 9:
9 = 3 × 3 = 3²

Prime factorization of 25:
25 = 5 × 5 = 5²

Now, let’s find the HCF and LCM:

HCF (Highest Common Factor):
We look for common prime factors raised to the lowest power. In the prime factorizations of 8 (2³), 9 (3²), and 25 (5²), there are no common prime factors. When there are no common prime factors, the HCF is 1.
HCF(8, 9, 25) = 1

LCM (Least Common Multiple):
We take the highest power of all prime factors that appear in any of the factorizations. The prime factors are 2, 3, and 5.
Highest power of 2 is 2³ (from 8).
Highest power of 3 is 3² (from 9).
Highest power of 5 is 5² (from 25).

LCM(8, 9, 25) = 2³ × 3² × 5²
LCM(8, 9, 25) = 8 × 9 × 25
LCM(8, 9, 25) = 72 × 25
LCM(8, 9, 25) = 1800

Therefore, the HCF of 8, 9, and 25 is 1, and the LCM of 8, 9, and 25 is 1800.

Question:

Prime factorization: Breaking down a number into its prime factors.
Highest Common Factor (HCF): The largest number that divides two or more numbers without leaving a remainder. It is found by taking the lowest power of common prime factors.
Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers. It is found by taking the highest power of all prime factors present in the numbers.
Verification: The product of two numbers is equal to the product of their HCF and LCM.


Step 1: Prime Factorization
Find the prime factorization of each number.
For 510:
510 = 2 × 255
255 = 3 × 85
85 = 5 × 17
So, 510 = 2¹ × 3¹ × 5¹ × 17¹

For 92:
92 = 2 × 46
46 = 2 × 23
So, 92 = 2² × 23¹

Step 2: Calculate HCF
The HCF is the product of the lowest powers of common prime factors.
The common prime factor is 2. The lowest power of 2 is 2¹.
HCF(510, 92) = 2¹ = 2

Step 3: Calculate LCM
The LCM is the product of the highest powers of all prime factors present in either number.
Prime factors are 2, 3, 5, 17, and 23.
Highest power of 2 is 2².
Highest power of 3 is 3¹.
Highest power of 5 is 5¹.
Highest power of 17 is 17¹.
Highest power of 23 is 23¹.
LCM(510, 92) = 2² × 3¹ × 5¹ × 17¹ × 23¹ = 4 × 3 × 5 × 17 × 23 = 12 × 5 × 17 × 23 = 60 × 17 × 23 = 1020 × 23 = 23460

Step 4: Verify the relationship LCM × HCF = product of the two numbers
Product of the two numbers = 510 × 92
510 × 92 = 46920

LCM × HCF = 23460 × 2
23460 × 2 = 46920

Since LCM × HCF = 46920 and the product of the numbers = 46920, the verification is successful.

Question:

Prime Factorisation: Breaking down a number into its prime factors.
HCF (Highest Common Factor): The largest number that divides into all the given numbers without leaving a remainder. It’s found by taking the lowest power of common prime factors.
LCM (Least Common Multiple): The smallest number that is a multiple of all the given numbers. It’s found by taking the highest power of all prime factors present in any of the numbers.


To find the LCM and HCF of 12, 15, and 21 using prime factorisation, we first break down each number into its prime factors.

1. Prime Factorisation:
12 = 2 × 2 × 3 = 2² × 3¹
15 = 3 × 5 = 3¹ × 5¹
21 = 3 × 7 = 3¹ × 7¹

2. HCF (Highest Common Factor):
To find the HCF, we look for prime factors that are common to all three numbers and take the lowest power of each common factor.
The common prime factor among 12, 15, and 21 is 3.
The lowest power of 3 present in all factorisations is 3¹.
Therefore, HCF(12, 15, 21) = 3.

3. LCM (Least Common Multiple):
To find the LCM, we take the highest power of all prime factors that appear in any of the factorisations.
The prime factors involved are 2, 3, 5, and 7.
The highest power of 2 is 2².
The highest power of 3 is 3¹.
The highest power of 5 is 5¹.
The highest power of 7 is 7¹.
Therefore, LCM(12, 15, 21) = 2² × 3¹ × 5¹ × 7¹ = 4 × 3 × 5 × 7 = 12 × 35 = 420.

So, the HCF of 12, 15, and 21 is 3, and the LCM of 12, 15, and 21 is 420.

Question:

Proof by contradiction. To prove a number is irrational, assume it is rational and show that this assumption leads to a contradiction.


We will prove that 3 + 2$\sqrt{5}$ is irrational by contradiction.

Step 1: Assume that 3 + 2$\sqrt{5}$ is rational.
If a number is rational, it can be expressed as a fraction p/q, where p and q are integers and q is not zero.
So, let 3 + 2$\sqrt{5}$ = p/q, where p, q $\in$ Z and q $\neq$ 0.

Step 2: Manipulate the equation to isolate the irrational part.
Subtract 3 from both sides of the equation:
2$\sqrt{5}$ = p/q – 3

Step 3: Combine the terms on the right side.
To subtract 3, we can write it as 3q/q:
2$\sqrt{5}$ = p/q – 3q/q
2$\sqrt{5}$ = (p – 3q) / q

Step 4: Isolate $\sqrt{5}$.
Divide both sides by 2:
$\sqrt{5}$ = (p – 3q) / (2q)

Step 5: Analyze the result and identify the contradiction.
The right side of the equation, (p – 3q) / (2q), is a ratio of two integers (p, 3q, and 2q are all integers since p and q are integers).
This means that the right side is a rational number.
Therefore, we have $\sqrt{5}$ = a rational number.

Step 6: State the contradiction.
We know that $\sqrt{5}$ is an irrational number.
Our assumption that 3 + 2$\sqrt{5}$ is rational has led us to the conclusion that $\sqrt{5}$ is rational, which is a contradiction.

Step 7: Conclude the proof.
Since our initial assumption leads to a contradiction, the assumption must be false.
Therefore, 3 + 2$\sqrt{5}$ is irrational.

Question:

Prime Factorization: Breaking down a number into its prime factors.
HCF (Highest Common Factor): The largest number that divides two or more numbers without leaving a remainder. It is found by multiplying the lowest powers of all common prime factors.
LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more numbers. It is found by multiplying the highest powers of all prime factors that appear in any of the numbers.
Verification: LCM × HCF = Product of the two numbers.


Step 1: Prime Factorization of the given numbers.
Find the prime factors of 26.
26 = 2 × 13

Find the prime factors of 91.
91 = 7 × 13

Step 2: Find the HCF.
Identify the common prime factors and their lowest powers.
The common prime factor is 13.
HCF(26, 91) = 13

Step 3: Find the LCM.
Identify all prime factors from both numbers and their highest powers.
The prime factors are 2, 7, and 13.
The highest power of 2 is 2¹.
The highest power of 7 is 7¹.
The highest power of 13 is 13¹.
LCM(26, 91) = 2 × 7 × 13 = 182

Step 4: Verify the relationship LCM × HCF = Product of the two numbers.
Calculate LCM × HCF:
182 × 13 = 2366

Calculate the Product of the two numbers:
26 × 91 = 2366

Step 5: Compare the results.
Since 2366 = 2366, the verification is successful.

Final Answer:
HCF(26, 91) = 13
LCM(26, 91) = 182
LCM × HCF = 182 × 13 = 2366
Product of the numbers = 26 × 91 = 2366
Therefore, LCM × HCF = Product of the two numbers.

Question:

Prime Factorization, HCF (Highest Common Factor), LCM (Least Common Multiple), Relationship between HCF, LCM and product of two numbers.


To find the HCF and LCM of 336 and 54, we first find their prime factorizations.

Prime factorization of 336:
336 = 2 × 168
168 = 2 × 84
84 = 2 × 42
42 = 2 × 21
21 = 3 × 7
So, 336 = 2 × 2 × 2 × 2 × 3 × 7 = 2⁴ × 3¹ × 7¹

Prime factorization of 54:
54 = 2 × 27
27 = 3 × 9
9 = 3 × 3
So, 54 = 2 × 3 × 3 × 3 = 2¹ × 3³

To find the HCF, we take the lowest power of common prime factors.
The common prime factors are 2 and 3.
Lowest power of 2 is 2¹.
Lowest power of 3 is 3¹.
HCF(336, 54) = 2¹ × 3¹ = 2 × 3 = 6

To find the LCM, we take the highest power of all prime factors present in either number.
Highest power of 2 is 2⁴.
Highest power of 3 is 3³.
Highest power of 7 is 7¹.
LCM(336, 54) = 2⁴ × 3³ × 7¹ = 16 × 27 × 7 = 432 × 7 = 3024

Now, we verify the relationship: LCM × HCF = product of the two numbers.
LCM × HCF = 3024 × 6 = 18144
Product of the two numbers = 336 × 54

Let’s calculate 336 × 54:
336 × 54 = 336 × (50 + 4)
= (336 × 50) + (336 × 4)
= 16800 + 1344
= 18144

So, LCM × HCF = 18144 and the product of the two numbers is also 18144.
Therefore, LCM × HCF = product of the two numbers is verified.

Answer:
HCF(336, 54) = 6
LCM(336, 54) = 3024
Verification: 3024 × 6 = 18144 and 336 × 54 = 18144. Thus, LCM × HCF = product of the two numbers.

Question:

Prime Factorisation: Breaking down a composite number into its prime number constituents.
LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more integers. For prime numbers, the LCM is their product.
HCF (Highest Common Factor): The largest positive integer that divides two or more integers without leaving a remainder. For distinct prime numbers, the HCF is 1.


The given numbers are 17, 23, and 29.
To find the HCF and LCM using the prime factorisation method, we first find the prime factors of each number.

17 is a prime number, so its prime factorisation is 17.
23 is a prime number, so its prime factorisation is 23.
29 is a prime number, so its prime factorisation is 29.

HCF (Highest Common Factor):
The HCF is the product of the lowest powers of all common prime factors.
In this case, there are no common prime factors among 17, 23, and 29. When there are no common prime factors, the HCF is 1.

LCM (Least Common Multiple):
The LCM is the product of the highest powers of all prime factors that appear in any of the factorisations.
The prime factors involved are 17, 23, and 29.
LCM(17, 23, 29) = 17 × 23 × 29

Calculating the LCM:
17 × 23 = 391
391 × 29 = 11339

Therefore, the HCF of 17, 23, and 29 is 1, and the LCM of 17, 23, and 29 is 11339.

Question:

The core concept needed here is proof by contradiction, specifically applied to irrational numbers. We will also use the property that if ‘a’ is a rational number and ‘b’ is an irrational number, then ‘ab’ is irrational. The definition of rational numbers (can be expressed as p/q where p and q are integers and q is not zero) and irrational numbers (cannot be expressed in this form) is also crucial.


We will prove that $7\sqrt{5}$ is irrational by assuming the opposite and showing it leads to a contradiction.
Step 1: Assume $7\sqrt{5}$ is rational.
Step 2: If $7\sqrt{5}$ is rational, it can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. So, $7\sqrt{5} = \frac{p}{q}$.
Step 3: Rearrange the equation to isolate $\sqrt{5}$. Divide both sides by 7: $\sqrt{5} = \frac{p}{7q}$.
Step 4: Analyze the right side of the equation. Since p and q are integers, 7q is also an integer. Therefore, $\frac{p}{7q}$ is a rational number.
Step 5: This implies that $\sqrt{5}$ is a rational number.
Step 6: However, it is a known fact that $\sqrt{5}$ is an irrational number. This is a contradiction.
Step 7: Since our initial assumption (that $7\sqrt{5}$ is rational) led to a contradiction, the assumption must be false.
Step 8: Therefore, $7\sqrt{5}$ is irrational.

Question:

The problem requires finding the least common multiple (LCM) of the time taken by Sonia and Ravi to complete one round. The LCM represents the smallest amount of time after which both will complete a whole number of rounds and meet at the starting point simultaneously.


Sonia takes 18 minutes for one round.
Ravi takes 12 minutes for one round.
They start at the same point and at the same time, and move in the same direction.
We need to find the time when they will meet again at the starting point. This means we are looking for a time that is a multiple of both 18 minutes and 12 minutes. The first time they will meet again at the starting point will be at the least common multiple (LCM) of their individual timings.

To find the LCM of 18 and 12, we can use prime factorization:
Prime factorization of 18 = 2 * 3 * 3 = 2^1 * 3^2
Prime factorization of 12 = 2 * 2 * 3 = 2^2 * 3^1

To find the LCM, we take the highest power of each prime factor present in the factorizations:
LCM(18, 12) = 2^2 * 3^2 = 4 * 9 = 36

Therefore, they will meet again at the starting point after 36 minutes.
At 36 minutes:
Sonia will have completed 36 / 18 = 2 rounds.
Ravi will have completed 36 / 12 = 3 rounds.
Both will be at the starting point.

Question:

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1 that have only two divisors: 1 and themselves (e.g., 2, 3, 5, 7, 11).


To express 140 as a product of its prime factors, we will repeatedly divide 140 by the smallest prime numbers until we are left with only prime numbers.

Step 1: Start with the smallest prime number, which is 2. Divide 140 by 2.
140 ÷ 2 = 70

Step 2: Now, take the result, 70, and divide it by the smallest prime number that divides it evenly. Again, this is 2.
70 ÷ 2 = 35

Step 3: Take the result, 35. Check if it’s divisible by 2. It’s not. Move to the next smallest prime number, which is 3. 35 is not divisible by 3. Move to the next smallest prime number, which is 5.
35 ÷ 5 = 7

Step 4: The result is 7. 7 is a prime number. So we stop here.

The prime factors of 140 are the numbers we divided by (2, 2, 5) and the final prime number (7).

Therefore, 140 can be expressed as the product of its prime factors: 2 × 2 × 5 × 7.

This can also be written using exponents: 2² × 5 × 7.

Question:

Prime factorization of numbers. A number ending in 0 must have both 2 and 5 as its prime factors.


We need to determine if 6^n can end with the digit 0 for any natural number n.

A natural number ends with the digit 0 if and only if it is divisible by 10.
The prime factorization of 10 is 2 * 5.
Therefore, for a number to end with the digit 0, its prime factorization must contain at least one factor of 2 and at least one factor of 5.

Let’s consider the prime factorization of 6^n.
We know that 6 = 2 * 3.
So, 6^n = (2 * 3)^n = 2^n * 3^n.

Now, let’s examine the prime factors of 6^n for any natural number n.
The prime factors of 6^n are only 2 and 3. There is no factor of 5 in the prime factorization of 6^n.

Since the prime factorization of 6^n does not contain the digit 5, it can never be divisible by 10.
Consequently, 6^n cannot end with the digit 0 for any natural number n.

For example:
If n = 1, 6^1 = 6 (ends in 6)
If n = 2, 6^2 = 36 (ends in 6)
If n = 3, 6^3 = 216 (ends in 6)

The last digit of 6^n will always be 6 for any natural number n.