Question

State the fundamental theorem of arithmetics.

Answer 1

The Fundamental Theorem of Arithmetic states that every composite number is a product of its primes and this factorisation is unique,apart from the order in which they occur.

Answer 2

Step-by-step explanation:

Fundamental Theorem of Arithmetic

Fundamental Theorem of Arithmetic states that every composite number can be expressed as the product of its prime factors. And, the prime factorisation is always unique irrespective of the order of the prime numbers.

Note: Fundamental Theorem of Arithmetic is used in finding the HCF and LCM of two or more numbers.

Example:

Find the HCF and LCM of 36, 44, and 60.

Solution:

36 can be expressed as:

36 = 2 × 2 × 3 × 3

= 22 × 32

44 = 2 × 2 × 11

= 22 × 11

60 = 2 × 2 × 3 × 5

= 22 × 3 × 5

LCM is the product of the greatest power of each prime factor involved in the number.

Here, four prime factors are involved in the numbers. They are 2, 3, 5 and 11.

For the prime factor 2, the highest power is 22.

For the prime factor 3, the highest power is 32.

For the prime factor 5, the highest power is 51.

For the prime factor 11, the highest power is 111.

Now, the product of these highest powers of each prime factor will give the LCM of the given numbers.

∴ LCM (36, 44, 60) = 22 × 32 × 5 × 11 = 1980

HCF is the product of the smallest power of each common prime factor in the numbers.

Here, the common prime factor involved in all three numbers is 2.

For the prime factor 2, the smallest common power is 22.

Now, the product of the smallest power of each common prime factor will give the HCF of the given numbers.

So, HCF (36, 44, 60) = 2 × 2 = 4