State the fundamental theorem of arithmetic class 10

Asked by admin @ 25/08/2021 in Math viewed by 297 People

State the fundamental theorem of Arithmetic
class 10

Answered by admin @ 25/08/2021

Fundamental Theorem of Arithmetic:

Given by given by Carl Friedrich Gauss, it states that every composite number can be written as the product of powers of primes E.g.: 30 = 2* 3* 5

Theorem 2 : Every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

The prime factorization of a natural number is unique, except for the order of its factors.

E.g. 30 = 2* 3* 5 = 2* 5 * 3 = 3 * 2 * 5 = 3 * 5 * 2 = 5 * 3 * 2 = 5 * 2 * 3

Prime factorization of 30 is unique; it has number 2, 3 & 5 ignoring the order.

Numerical: Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.

Solution: If the number 4n, for any n, were to end with the digit zero, then it would be divisible by 5. But as per factorization theorem, 4n has the only prime factor 2 , so it is not divisible by 10.

Numerical: Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Solution: We have : 6 = 21 × 31 and 20 = 2 × 2 × 5 = 22 × 51.

Note that HCF (6, 20) = 21 = 2

HCS is Product of the smallest power of each common prime factor in the numbers.

LCM (6, 20) = 22 × 31 × 51 = 60

LCM is Product of the greatest power of each prime factor, involved in the numbers.

State the fundamental theorem of arithmetic class 10

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