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# Fundamental Theorem of Arithmetic

Every natural number except 1 can be factorized as a product of primes and this factorization is unique except for the order in which the prime factors are written.

Fundamental Theorem of Arithmetic

Let us consider the following conversation between a Teacher and students.

Teacher : Factorise the number 240.

Malar : 24 ×10

Raghu : 8×30

Iniya : 12×20

Kumar : 15×16

Malar : Whose answer is correct Sir?

Teacher : All the answers are correct.

Raghu : How sir?

Teacher : Split each of the factors into product of prime numbers.

Malar : 2×2×2×3×2×5

Raghu : 2×2×2×2×3×5

Iniya : 2×2×3×2×2×5

Kumar : 3×5×2×2×2×2

Teacher : Good! Now, count the number of 2’s, 3’s and 5’s.

Malar : I got four 2’s, one 3 and one 5.

Raghu : I got four 2’s, one 3 and one 5.

Iniya : I also got the same numbers too.

Kumar : Me too sir.

Malar : All of us got four 2’s, one 3 and one 5. This is very surprising to us.

Teacher : Yes, It should be. Once any number is factorized up to a product of prime numbers, everyone should get the same collection of prime numbers.

This concept leads us to the following important theorem.

## Theorem 4 (Fundamental Theorem of Arithmetic) (without proof)

“Every natural number except 1 can be factorized as a product of primes and this factorization is unique except for the order in which the prime factors are written.”

The fundamental theorem asserts that every composite number can be decomposed as a product of prime numbers and that the decomposition is unique. In the sense that there is one and only way to express the decomposition as product of primes. In general, we conclude that given a composite number N, we decompose it uniquely in the form = p1q1  × p2q2  × p3q3  × …… × pnqn   where p1, p2, p3,……. ,pn are primes and q1, q2, q3, …. qn are natural.

First, we try to factorize N into its factors. If all the factors are themselves primes then we can stop. Otherwise, we try to further split the factors which are not prime. Continue the process till we get only prime numbers.

## Illustration

For example, if we try to factorize32760 we get

32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 ×13

= 23 × 32 × 51 × 71 ×131

Thus, in whatever way we try to factorize 32760, we should finally get three 2’s, two 3’s, one 5, one 7 and one 13.

The fact that “Every composite number can be written uniquely as the product of power of primes” is called Fundamental Theorem of Arithmetic.

## Significance of the Fundamental Theorem of Arithmetic

The fundamental theorem about natural numbers except 1, that we  have  stated above has several applications, both in Mathematics and in other fields. The theorem is vastly important in Mathematics, since it highlights the fact that prime numbers are the ‘Building Blocks’ for all the positive integers. Thus, prime numbers can be compared to atoms making up a molecule.

1. If a prime number p divides ab then either p divides a or p divides b.

That is p divides at least one of them.

2. If a composite number n divides ab, then n neighter divide a nor b.

For example, 6 divides 4 × 3 but 6 neither divide 4 nor 3.

### Example 2.7

In the given factor tree, find the numbers m and n.

### Solution

Value of the first box from bottom = 5 × 2 = 10

Value of n = 5 ×10 = 50

Value of the second box from bottom = 3 × 50 = 150

Value of m = 2 ×150 = 300

Thus, the required numbers are m = 300, n = 50 ### Example 2.8

Can the number 6n , n being a natural number end with the digit 5? Give reason for your answer.

### Solution

Since 6n= (2 × 3)n = 2n × 3n

2 is a factor of 6n . So, 6n is always even. But any number whose last digit is 5 is always odd. Hence, 6n cannot end with the digit 5

Example 2.9

Is 7 × 5 × 3 × 2 + 3 a composite number? Justify your answer.

Solution

Yes, the given number is a composite number, because

7 × 5 × 3 × 2 + 3 = 3 × (7 × 5 × 2 + 1) = 3 × 71

Since the given number can be factorized in terms of two primes, it is a composite number.

### Example 2.10

a’ and ‘b’ are two positive integers such that ab ×ba = 800. Find ‘a’ and ‘b’.

Solution The number 800 can be factorized as

800 = 2 × 2 × 2 × 2 × 2 × 5 × 5 = 25 × 52

Hence, ab ×ba = 25 × 52

This implies that a = 2 and b = 5 (or) a = 5 and b = 2

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