REAL NUMBERS - Q&A
EXERCISE 1.11. Express each number as a product of its prime factors:
(i) 140
(ii) 156
(iii) 3825
(iv) 5005
(v) 7429
Solution:
To find the prime factors, we divide the numbers by prime numbers (2, 3, 5, 7, 11, etc.) successively.
(i) 140
140 ÷ 2 = 70
70 ÷ 2 = 35
35 ÷ 5 = 7
7 ÷ 7 = 1
Therefore, 140 = 2 × 2 × 5 × 7 = 22 × 5 × 7
(ii) 156
156 ÷ 2 = 78
78 ÷ 2 = 39
39 ÷ 3 = 13
13 ÷ 13 = 1
Therefore, 156 = 2 × 2 × 3 × 13 = 22 × 3 × 13
(iii) 3825
(Sum of digits = 3+8+2+5 = 18, so it is divisible by 3)
3825 ÷ 3 = 1275
1275 ÷ 3 = 425
425 ÷ 5 = 85
85 ÷ 5 = 17
17 ÷ 17 = 1
Therefore, 3825 = 3 × 3 × 5 × 5 × 17 = 32 × 52 × 17
(iv) 5005
5005 ÷ 5 = 1001
1001 ÷ 7 = 143
143 ÷ 11 = 13
13 ÷ 13 = 1
Therefore, 5005 = 5 × 7 × 11 × 13
(v) 7429
(We check for divisibility by primes like 2, 3, 5, 7, etc. It is divisible by 17)
7429 ÷ 17 = 437
437 ÷ 19 = 23
23 ÷ 23 = 1
Therefore, 7429 = 17 × 19 × 23
2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(i) 26 and 91
(ii) 510 and 92
(iii) 336 and 54
Solution:
(i) 26 and 91
Step 1: Find Prime Factors.
26 = 2 × 13
91 = 7 × 13
Step 2: Find HCF and LCM.
HCF (product of smallest power of each common prime factor) = 13
LCM (product of greatest power of each prime factor) = 2 × 7 × 13 = 182
Step 3: Verify LCM × HCF = Product of numbers.
LCM × HCF = 182 × 13 = 2366
Product of numbers = 26 × 91 = 2366
Hence, Verified.
(ii) 510 and 92
Step 1: Find Prime Factors.
510 = 2 × 3 × 5 × 17
92 = 2 × 2 × 23 = 22 × 23
Step 2: Find HCF and LCM.
HCF (common factor is 2) = 2
LCM (22 × 3 × 5 × 17 × 23) = 4 × 3 × 5 × 17 × 23 = 23460
Step 3: Verify LCM × HCF = Product of numbers.
LCM × HCF = 23460 × 2 = 46920
Product of numbers = 510 × 92 = 46920
Hence, Verified.
(iii) 336 and 54
Step 1: Find Prime Factors.
336 = 2 × 2 × 2 × 2 × 3 × 7 = 24 × 3 × 7
54 = 2 × 3 × 3 × 3 = 2 × 33
Step 2: Find HCF and LCM.
HCF (Smallest power of common factors 2 and 3) = 21 × 31 = 6
LCM (Greatest power of factors 2, 3, and 7) = 24 × 33 × 7 = 16 × 27 × 7 = 3024
Step 3: Verify LCM × HCF = Product of numbers.
LCM × HCF = 3024 × 6 = 18144
Product of numbers = 336 × 54 = 18144
Hence, Verified.
3. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) 12, 15 and 21
(ii) 17, 23 and 29
(iii) 8, 9 and 25
Solution:
(i) 12, 15 and 21
12 = 22 × 3
15 = 3 × 5
21 = 3 × 7
HCF (Common factor is 3) = 3
LCM (22 × 3 × 5 × 7) = 4 × 3 × 5 × 7 = 420
(ii) 17, 23 and 29
17 = 17 × 1
23 = 23 × 1
29 = 29 × 1
(Since they are all prime numbers)
HCF = 1
LCM = 17 × 23 × 29 = 11339
(iii) 8, 9 and 25
8 = 23
9 = 32
25 = 52
There are no common factors other than 1.
HCF = 1
LCM = 23 × 32 × 52 = 8 × 9 × 25 = 1800
4. Given that HCF (306, 657) = 9 find LCM (306, 657).
Solution:
We know the property: LCM × HCF = Product of the two numbers.
Let the two numbers be a and b.
Here, a = 306, b = 657, and HCF = 9.
LCM × 9 = 306 × 657
LCM = (306 × 657) / 9
LCM = 34 × 657
LCM = 22338
5. Check whether 6n can end with the digit 0 for any natural number n.
Solution:
For a number to end with the digit 0, it must be divisible by 10. This means it must have both 2 and 5 as its prime factors.
Let us look at the prime factorization of 6n.
6n = (2 × 3)n = 2n × 3n
In the prime factorization of 6n, the only prime factors are 2 and 3.
According to the Fundamental Theorem of Arithmetic, this factorization is unique.
Since the prime factor 5 is missing, 6n cannot be divisible by 5.
Therefore, 6n can never end with the digit 0 for any natural number n.
6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Solution:
A composite number is a number that has factors other than 1 and itself.
Case 1: 7 × 11 × 13 + 13
We can take 13 common from the expression.
= 13 × (7 × 11 + 1)
= 13 × (77 + 1)
= 13 × 78
Since this number can be expressed as a product of two integers (13 and 78) other than 1, it is a composite number.
Case 2: 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
We can take 5 common from the expression.
= 5 × (7 × 6 × 4 × 3 × 2 × 1 + 1)
= 5 × (1008 + 1)
= 5 × 1009
Since this number can be expressed as a product of 5 and 1009, it has factors other than 1 and itself. Therefore, it is a composite number.
7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
Solution:
To find when they will meet again at the starting point, we need to find a time that is a common multiple of both 18 minutes and 12 minutes. The earliest they will meet is the Lowest Common Multiple (LCM) of 18 and 12.
Prime factorization of 18: 2 × 3 × 3 = 2 × 32
Prime factorization of 12: 2 × 2 × 3 = 22 × 3
LCM (18, 12) = Product of greatest power of each prime factor.
LCM = 22 × 32
LCM = 4 × 9
LCM = 36
Therefore, Sonia and Ravi will meet again at the starting point after 36 minutes.
EXERCISE 1.2
1. Prove that √5 is irrational.
Solution:
We will prove this by the method of contradiction.
Assumption: Let us assume, to the contrary, that √5 is rational.
If it is rational, we can find integers a and b (where b ≠ 0) such that √5 = a / b.
We also assume that a and b are coprime (i.e., they have no common factor other than 1).
So, √5 = a / b
⇒ b√5 = a
Squaring both sides:
5b2 = a2 ... (Equation 1)
This means a2 is divisible by 5. By the theorem (if a prime p divides a2, then p divides a), it follows that a is divisible by 5.
So we can write a = 5c for some integer c.
Substitute a = 5c into Equation 1:
5b2 = (5c)2
5b2 = 25c2
Dividing both sides by 5:
b2 = 5c2
This means b2 is divisible by 5. Consequently, b is divisible by 5.
Conclusion: We have shown that a and b both have 5 as a common factor.
But this contradicts our initial assumption that a and b are coprime (have no common factors).
This contradiction has arisen because of our incorrect assumption that √5 is rational.
Therefore, √5 is irrational.
2. Prove that 3 + 2√5 is irrational.
Solution:
We use the method of contradiction.
Assumption: Let us assume that 3 + 2√5 is rational.
Then we can find co-prime integers a and b (b ≠ 0) such that:
3 + 2√5 = a / b
Rearranging the equation to isolate √5:
2√5 = (a / b) - 3
2√5 = (a - 3b) / b
√5 = (a - 3b) / 2b
Since a and b are integers, (a - 3b) and 2b are also integers.
Therefore, the Right Hand Side (RHS), (a - 3b) / 2b, is a rational number.
This implies that the Left Hand Side, √5, must also be rational.
Contradiction: But we know that √5 is irrational.
This contradicts the fact that √5 is irrational.
This contradiction has arisen because of our incorrect assumption that 3 + 2√5 is rational.
Therefore, 3 + 2√5 is irrational.
3. Prove that the following are irrationals:
(i) 1 / √2
(ii) 7√5
(iii) 6 + √2
Solution:
(i) 1 / √2
Assume 1 / √2 is rational.
Then 1 / √2 = a / b, where a, b are coprime integers and b ≠ 0.
Taking the reciprocal of both sides:
√2 = b / a
Since a and b are integers, b / a is rational.
This implies √2 is rational, which contradicts the fact that √2 is irrational.
Therefore, 1 / √2 is irrational.
(ii) 7√5
Assume 7√5 is rational.
Then 7√5 = a / b for coprime integers a, b.
√5 = a / 7b
Since a, b, and 7 are integers, a / 7b is rational.
This implies √5 is rational, which contradicts the known fact that √5 is irrational.
Therefore, 7√5 is irrational.
(iii) 6 + √2
Assume 6 + √2 is rational.
Then 6 + √2 = a / b for coprime integers a, b.
√2 = (a / b) - 6
√2 = (a - 6b) / b
Since a and b are integers, (a - 6b) / b is rational.
This implies √2 is rational, which contradicts the known fact that √2 is irrational.
Therefore, 6 + √2 is irrational.
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Quick Review Flashcards - Click to flip and test your knowledge!
Question
What is the primary use of Euclid's division algorithm as discussed in the text?
Answer
To compute the Highest Common Factor ($HCF$) of two positive integers.
Question
State the Fundamental Theorem of Arithmetic.
Answer
Every composite number can be expressed as a unique product of primes, regardless of the order of factors.
Question
Which property of integers does Euclid's division algorithm primarily deal with?
Answer
The divisibility of integers.
Question
According to the Fundamental Theorem of Arithmetic, the factorisation of a composite number is unique except for which aspect?
Answer
The order in which the prime factors occur.
Question
What is revealed by looking at the prime factorisation of the denominator $q$ of a rational number $\frac{p}{q}$?
Answer
Whether the decimal expansion is terminating or non-terminating repeating.
Question
Concept: Composite Number
Answer
Definition: A positive integer that can be expressed as a product of prime numbers in a unique way.
Question
What determines if the decimal expansion of a rational number $\frac{p}{q}$ ($q \neq 0$) terminates?
Answer
The prime factorisation of the denominator $q$.
Question
Who provided the first correct proof of the Fundamental Theorem of Arithmetic in his work 'Disquisitiones Arithmeticae'?
Answer
Carl Friedrich Gauss.
Question
Which historical mathematician is often referred to as the 'Prince of Mathematicians'?
Answer
Carl Friedrich Gauss.
Question
The Fundamental Theorem of Arithmetic was likely first recorded as Proposition 14 of Book IX in which historical text?
Answer
Euclid’s Elements.
Question
In the standard form of prime factorisation $x = p_1 p_2 \dots p_n$, how are the primes usually ordered?
Answer
In ascending order ($p_1 \leq p_2 \leq \dots \leq p_n$).
Question
Why can the number $4^n$ never end with the digit zero for any natural number $n$?
Answer
The only prime factor in $4^n$ is $2$, and it lacks the prime factor $5$ required to end in zero.
Question
Term: Prime Factorisation Method
Answer
Definition: A method used to find the $HCF$ and $LCM$ of positive integers by expressing them as products of prime powers.
Question
How is the $HCF$ of two numbers calculated using prime factorisation?
Answer
By taking the product of the smallest power of each common prime factor in the numbers.
Question
How is the $LCM$ of two numbers calculated using prime factorisation?
Answer
By taking the product of the greatest power of each prime factor involved in the numbers.
Question
For any two positive integers $a$ and $b$, what is the relationship between their $HCF$, $LCM$, and their product?
Answer
$HCF(a, b) \times LCM(a, b) = a \times b$.
Question
Does the formula $HCF(a, b, c) \times LCM(a, b, c) = a \times b \times c$ hold true for three numbers?
Answer
No, the product of three numbers is generally not equal to the product of their $HCF$ and $LCM$.
Question
Find the $HCF$ of $96$ and $404$ using prime factorisation.
Answer
$4$ (since $96 = 2^5 \times 3$ and $404 = 2^2 \times 101$).
Question
Given $HCF(306, 657) = 9$, how would you find $LCM(306, 657)$?
Answer
By calculating $\frac{306 \times 657}{9}$.
Question
Theorem: If a prime $p$ divides $a^2$, then $p$ must also divide _____.
Answer
$a$ (where $a$ is a positive integer).
Question
What specific technique is used in the text to prove that $\sqrt{2}$ is irrational?
Answer
Proof by contradiction.
Question
When assuming $\sqrt{2} = \frac{a}{b}$ to prove irrationality, what property is assumed for $a$ and $b$?
Answer
That $a$ and $b$ are coprime (they have no common factors other than $1$).
Question
If $2$ divides $a^2$ and $a$ is an integer, what is the conclusion regarding $a$ based on Theorem 1.2?
Answer
$2$ divides $a$.
Question
State the definition of an irrational number.
Answer
A number that cannot be written in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
Question
What is the result of adding or subtracting a rational number and an irrational number?
Answer
The result is an irrational number.
Question
What is the result of the product of a non-zero rational number and an irrational number?
Answer
The result is an irrational number.
Question
Prove $3\sqrt{2}$ is irrational by contradiction: if $3\sqrt{2} = \frac{a}{b}$, what does $\sqrt{2}$ equal?
Answer
$\sqrt{2} = \frac{a}{3b}$.
Question
Why is $3\sqrt{2} = \frac{a}{3b}$ a contradiction if $a, b,$ and $3$ are integers?
Answer
It implies $\sqrt{2}$ is rational, which contradicts the fact that $\sqrt{2}$ is irrational.
Question
Express $140$ as a product of its prime factors.
Answer
$2^2 \times 5 \times 7$.
Question
Express $156$ as a product of its prime factors.
Answer
$2^2 \times 3 \times 13$.
Question
Express $3825$ as a product of its prime factors.
Answer
$3^2 \times 5^2 \times 17$.
Question
Express $5005$ as a product of its prime factors.
Answer
$5 \times 7 \times 11 \times 13$.
Question
Express $7429$ as a product of its prime factors.
Answer
$17 \times 19 \times 23$.
Question
Why is $7 \times 11 \times 13 + 13$ a composite number?
Answer
It can be factorised as $13(7 \times 11 + 1)$, showing it has factors other than $1$ and itself.
Question
Formula: $LCM(p, q, r) = \dots$ (using $HCF$)
Answer
$\frac{p \cdot q \cdot r \cdot HCF(p, q, r)}{HCF(p, q) \cdot HCF(q, r) \cdot HCF(p, r)}$.
Question
Formula: $HCF(p, q, r) = \dots$ (using $LCM$)
Answer
$\frac{p \cdot q \cdot r \cdot LCM(p, q, r)}{LCM(p, q) \cdot LCM(q, r) \cdot LCM(p, r)}$.
Question
In the circular path problem, Sonia takes $18$ mins and Ravi takes $12$ mins. To find when they meet at the start, what must be calculated?
Answer
The $LCM$ of $18$ and $12$.
Question
What is the $LCM$ of $18$ and $12$?
Answer
$36$.
Question
Which property of prime factorisation ensures that no natural number $n$ exists such that $6^n$ ends with the digit zero?
Answer
Uniqueness of the Fundamental Theorem of Arithmetic (prime factors are $2$ and $3$, but $5$ is missing).
Question
What are the only prime factors of $a^2$ if the prime factorisation of $a$ is $p_1 p_2 \dots p_n$?
Answer
$p_1, p_2, \dots, p_n$.
Question
How is the $HCF(6, 72, 120)$ found using the prime factorisation method?
Answer
$HCF = 2^1 \times 3^1 = 6$ (product of smallest powers of common prime factors).
Question
How is the $LCM(6, 72, 120)$ found using the prime factorisation method?
Answer
$LCM = 2^3 \times 3^2 \times 5^1 = 360$ (product of greatest powers of all prime factors involved).
Question
If $a$ and $b$ have at least $3$ as a common factor, why does this contradict the assumption that they are coprime?
Answer
Coprime numbers, by definition, have no common factors other than $1$.
Question
Identify the context of the statement: 'Every natural number can be written as a product of its prime factors.'
Answer
The Fundamental Theorem of Arithmetic.
Question
What is the $HCF$ of the integers $17, 23,$ and $29$?
Answer
$1$ (as all three are prime numbers).
Question
What is the $LCM$ of the integers $8, 9,$ and $25$?
Answer
$1800$ (calculated as $2^3 \times 3^2 \times 5^2$).
Question
What mathematical tool is often used to visualize the prime factorisation of a large number like $32760$?
Answer
A factor tree.
Question
Under what condition can we say that the way a number is factorised is completely unique?
Answer
When the prime factors are written in a specific order, such as ascending order.
Question
What determines if a prime $p$ is a factor of $a$ if it is known that $p$ is a factor of $a^2$?
Answer
The uniqueness part of the Fundamental Theorem of Arithmetic.
Question
Is $\pi - 3$ considered rational or irrational?
Answer
Irrational.
Question
If $b^2 = 3c^2$, what can be concluded about the divisibility of $b^2$ by $3$?
Answer
$b^2$ is divisible by $3$.
Question
Is the number $0.10110111011110 \dots$ rational or irrational?
Answer
Irrational (it is non-terminating and non-repeating).