Question

Find HCF of $70$ and $245$ using Fundamental Theorem of Arithmetic.
A) 35
B) 30
C) 15
D) 25

Answer 1

Hint: The fundamental theorem of arithmetic is also called a unique factorization theorem, which states that every composite number can be expressed as a product of prime numbers meaning the product of those numbers or a single which is/are common in both the numbers and is highest amongst all the factors present. That highest common factor/s results to form HCF.

Complete step-by-step answer:
Given numbers are \[70\] and \[245\].
The fundamental theorem of arithmetic is also called the unique factorization theorem, which states that every composite number can be expressed as a product of prime numbers.
The number \[70\] can be written as a product of prime factors as,
\[70 = 2\times 5\times 7\]
The number 245 can be written as a product of prime factors as,
\[245 = 5\times 7\times 7\]
Now, HCF of any two numbers is the highest common factor to the given numbers. The common factors of \[70\] and \[245\] are \[5\] and \[7\].
So, the HCF is
\[5\times 7 = 35\]

Hence the correct option is (A).

Note: The LCM of two numbers can also be evaluated using the fundamental theorem of arithmetic, for LCM the result will be a product of all the numbers/factors of the two numbers for both common and uncommon.