1.2-The Fundamental Theorem of Arithmetic

1.2-The Fundamental Theorem of Arithmetic Important Formulae

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Grade 10 → Math → Real Numbers → 1.2-The Fundamental Theorem of Arithmetic

After successful completion of this topic, you should be able to:

  • Use the Fundamental Theorem of Arithmetic in order to calculate HCF and LCM of the given numbers in the context of the given problem.

The Fundamental Theorem of Arithmetic is a crucial concept in number theory, which states that every integer greater than 1 can be uniquely expressed as a product of prime numbers, up to the order of the factors. This theorem lays the foundation for various topics in mathematics, including prime factorization, divisibility, and number systems.

1. Definition of Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In contrast, a composite number has factors other than 1 and itself. The first few prime numbers are:

2, 3, 5, 7 ,11, 13, 17, 19,....

2. Prime Factorization

Prime factorization is the process of expressing a composite number as a product of its prime factors. For example, the prime factorization of 60 can be expressed as:

$60 = 2^2 \times 3^1 \times 5^1$

3. Uniqueness of Prime Factorization

The uniqueness aspect of the Fundamental Theorem of Arithmetic states that while a number can be expressed as a product of primes in different orders, the actual prime factors and their respective powers remain constant. For instance:

Both $12$ and $12$ can be factored as:

$12 = 2^2 \times 3^1$

or

$12 = 3^1 \times 2^2$

This shows that no matter the order of multiplication, the prime factorization of 12 is always $2^2 \times 3^1$.

4. Application of the Theorem

The Fundamental Theorem of Arithmetic has several applications, including:

  • Finding the Greatest Common Divisor (GCD): The GCD of two numbers can be determined by taking the product of the smallest powers of common prime factors.
  • Finding the Least Common Multiple (LCM): The LCM can be found by taking the product of the highest powers of all prime factors involved.
  • Solving Diophantine Equations: It helps in finding integer solutions to equations involving multiple variables.
5. Examples

Let’s consider some examples to illustrate the theorem:

  • For the number $18$, the prime factorization is:

    • $18 = 2^1 \times 3^2$

  • For the number $45$, the prime factorization is:

    • $45 = 3^2 \times 5^1$

When finding the GCD of $18$ and $45$:

  • Prime factorization of $18$: $2^1 \times 3^2$
  • Prime factorization of $45$: $3^2 \times 5^1$

The common prime factor is $3$ with the minimum power of $3^2$, hence:

GCD($18, 45) = 3^2 = 9$.

6. Importance of the Theorem

The Fundamental Theorem of Arithmetic emphasizes the significance of prime numbers in the structure of integers. It serves as a cornerstone for further studies in algebra, number theory, and beyond, illustrating how numbers can be constructed from their fundamental building blocks.


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Express each number as a product of its prime factors:

(i) 140
(ii) 156
(iii) 3825
(iv) 5005
(v) 7429

Solution:

Express each number as a product of its prime factors:
(i) 140 = $2 \times 2 \times 5 \times 7$
(ii) 156 = $2 \times 2 \times 3 \times 13$
(iii) 3825 = $3 \times 3 \times 5 \times 5 \times 17$
(iv) 5005 = $5 \times 7 \times 11 \times 13$
(v) 7429 = $17 \times 19 \times 23$

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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:

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
HCF of 26 and 91: Prime factorization of 26 = $2 \times 13$
Prime factorization of 91 = $7 \times 13$
Common factors = 13
Thus, $HCF = 13$
LCM of 26 and 91: LCM = Product of the highest powers of all prime factors = $2 \times 7 \times 13 = 182$
Verification: $LCM \times HCF = 182 \times 13 = 2366$
Product of the two numbers = $26 \times 91 = 2366$
Thus, $LCM \times HCF = 26 \times 91$

(ii) 510 and 92
HCF of 510 and 92: Prime factorization of 510 = $2 \times 3 \times 5 \times 17$
Prime factorization of 92 = $2^2 \times 23$
Common factor = 2
Thus, $HCF = 2$
LCM of 510 and 92: LCM = Product of the highest powers of all prime factors = $2^2 \times 3 \times 5 \times 17 \times 23 = 2^2 \times 3 \times 5 \times 17 \times 23 = 3460$
Verification: $LCM \times HCF = 3460 \times 2 = 6920$
Product of the two numbers = $510 \times 92 = 6920$
Thus, $LCM \times HCF = 510 \times 92$

(iii) 336 and 54
HCF of 336 and 54: Prime factorization of 336 = $2^4 \times 3 \times 7$
Prime factorization of 54 = $2 \times 3^3$
Common factors = $2 \times 3$
Thus, $HCF = 6$
LCM of 336 and 54: LCM = Product of the highest powers of all prime factors = $2^4 \times 3^3 \times 7 = 3024$
Verification: $LCM \times HCF = 3024 \times 6 = 18144$
Product of the two numbers = $336 \times 54 = 18144$
Thus, $LCM \times HCF = 336 \times 54$

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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:

Find the LCM and HCF of the following integers by applying the prime factorisation method:
(i) 12, 15 and 21

Prime factorisation of 12 = $2^2 \times 3$

Prime factorisation of 15 = $3 \times 5$

Prime factorisation of 21 = $3 \times 7$

HCF = Common prime factors with lowest powers = $3$

LCM = Product of highest powers of all prime factors = $2^2 \times 3 \times 5 \times 7 = 420$

(ii) 17, 23 and 29

Prime factorisation of 17 = $17$ (prime number)

Prime factorisation of 23 = $23$ (prime number)

Prime factorisation of 29 = $29$ (prime number)

HCF = No common prime factors = $1$

LCM = Product of all prime factors = $17 \times 23 \times 29 = 11251$

(iii) 8, 9 and 25

Prime factorisation of 8 = $2^3$

Prime factorisation of 9 = $3^2$

Prime factorisation of 25 = $5^2$

HCF = No common prime factors = $1$

LCM = Product of highest powers of all prime factors = $2^3 \times 3^2 \times 5^2 = 900$

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Given that HCF (306, 657) = 9, find LCM (306, 657).

Solution:

Given that HCF (306, 657) = 9, find LCM (306, 657).

We know that,

HCF × LCM = Product of the two numbers

So, we have:

HCF (306, 657) = 9

Product of 306 and 657 = 306 × 657

Now, using the formula:

9 × LCM (306, 657) = 306 × 657

LCM (306, 657) = (306 × 657) / 9

First, calculate the product of 306 and 657:

306 × 657 = 201,762

Now, divide the product by 9:

LCM (306, 657) = 201,762 / 9 = 22,419

The LCM of 306 and 657 is 22,419.

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Check whether $6^n$ can end with the digit 0 for any natural number n.

Solution:

Question: Check whether $6^n$ can end with the digit 0 for any natural number n.

We are tasked with determining whether $6^n$ can end with the digit 0 for any natural number $n$.

To solve this, we need to analyze the units digit of $6^n$ for various values of $n$.

Let's calculate the first few powers of 6:

When $n = 1$, $6^1 = 6$ (units digit is 6).

When $n = 2$, $6^2 = 36$ (units digit is 6).

When $n = 3$, $6^3 = 216$ (units digit is 6).

When $n = 4$, $6^4 = 1296$ (units digit is 6).

It can be observed that for all natural numbers $n$, the units digit of $6^n$ is always 6.

Since 6 does not end in 0, $6^n$ can never end with the digit 0 for any natural number $n$.

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Explain why 7×11×13+13 and 7×6×5×4×3×2×1+5 are composite numbers.

Solution:

Explanation of Why 7×11×13+13 and 7×6×5×4×3×2×1+5 are Composite Numbers

To prove that 7×11×13+13 and 7×6×5×4×3×2×1+5 are composite numbers, we will check if they can be factored into smaller integers.

1. 7×11×13 + 13

We can factor out 13 from the expression:

$7×11×13 + 13 = 13(7×11 + 1)$

Now, simplify the expression inside the parentheses:

$7×11 + 1 = 77 + 1 = 78$

So, the expression becomes:

$13 × 78$

Since this number is the product of 13 and 78, both of which are greater than 1, it is a composite number.

2. 7×6×5×4×3×2×1 + 5

We first calculate the value of $7×6×5×4×3×2×1$:

$7×6×5×4×3×2×1 = 5040$

Now, add 5 to this value:

$5040 + 5 = 5045$

We need to check if 5045 is composite. Let's check if it can be factored:

5045 is divisible by 5 (since it ends in 5), so we divide:

$5045 ÷ 5 = 1009$

Thus, 5045 = 5 × 1009. Since 5045 is the product of 5 and 1009, both of which are greater than 1, it is a composite number.

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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:

Question:

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 solve this, we need to find the least common multiple (LCM) of 18 and 12, as that will tell us when both Sonia and Ravi will be at the starting point again at the same time.

The prime factorization of 18 is: $18 = 2 \times 3^2$

The prime factorization of 12 is: $12 = 2^2 \times 3$

Now, the LCM is found by taking the highest powers of all prime factors that appear in the factorizations.

The LCM of 18 and 12 is: $LCM(18, 12) = 2^2 \times 3^2 = 36$

So, Sonia and Ravi will meet again at the starting point after $36$ minutes.

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