Notes On Fundamental Theorem of Arithmetic - CBSE Class 10 Maths
Fundamental Theorem of Arithmetic: Fundamental Theorem of Arithmetic states that every composite number greater than 1 can be expressed or factorised as a unique product of prime numbers except in the order of the prime factors. We can write the prime factorisation of a number in the form of powers of its prime factors. By expressing any two numbers as their prime factors, their highest common factor (HCF) and lowest common multiple (LCM) can be easily calculated. The HCF of two numbers is equal to the product of the terms containing the least powers of common prime factors of the two numbers. The LCM of two numbers is equal to the product of the terms containing the greatest powers of all prime factors of the two numbers. For any two positive integers a and b, HCF(a , b) x LCM(a , b) = a x b. For any three positive integers a, b and c, LCM(a, b, c) = $\frac{\text{a.b.c HCF(a, b, c)}}{\text{HCF(a, b).HCF(b, c).HCF(c, a)}}$ HCF(a, b, c) = $\frac{\text{a.b.c LCM(a, b, c)}}{\text{LCM(a, b).LCM(b, c).LCM(c, a)}}$. Note: The product of the given numbers is equal to the product of their HCF and LCM. This result is true for all positive integers and is often used to find the HCF of two given numbers if their LCM is given and vice versa.

#### Summary

Fundamental Theorem of Arithmetic: Fundamental Theorem of Arithmetic states that every composite number greater than 1 can be expressed or factorised as a unique product of prime numbers except in the order of the prime factors. We can write the prime factorisation of a number in the form of powers of its prime factors. By expressing any two numbers as their prime factors, their highest common factor (HCF) and lowest common multiple (LCM) can be easily calculated. The HCF of two numbers is equal to the product of the terms containing the least powers of common prime factors of the two numbers. The LCM of two numbers is equal to the product of the terms containing the greatest powers of all prime factors of the two numbers. For any two positive integers a and b, HCF(a , b) x LCM(a , b) = a x b. For any three positive integers a, b and c, LCM(a, b, c) = $\frac{\text{a.b.c HCF(a, b, c)}}{\text{HCF(a, b).HCF(b, c).HCF(c, a)}}$ HCF(a, b, c) = $\frac{\text{a.b.c LCM(a, b, c)}}{\text{LCM(a, b).LCM(b, c).LCM(c, a)}}$. Note: The product of the given numbers is equal to the product of their HCF and LCM. This result is true for all positive integers and is often used to find the HCF of two given numbers if their LCM is given and vice versa.

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