Question

What is a HCF?
(a) The highest common factor (HCF) of two numbers (or expressions) is the largest number or expression which is a factor of both.
(b) The highest common factor (HCF) of two numbers (or expressions) is the smallest number or expression which is a factor of both.
(c) The highest common factor (HCF) of two numbers (or expressions) is the largest factor of either of the two numbers (or expression).
(d) The highest common factor (HCF) of two numbers (or expressions) is the smallest factor of either of the two numbers (or expression).

Answer 1

Hint: To solve the problem, we should know the basic concepts of the highest common factor (HCF). Basically, we try to find the number of divisors of two numbers and then we find the largest divisor to find the highest common factor (HCF).

Complete step-by-step solution -
Basically, we try to find the basics of the highest common factor (HCF) first before we can arrive at the correct answer to the question given above. In simple terms, the highest common factor (HCF) is the largest divisor of any 2 numbers. For example, in case of 12 and 18, the divisors of these numbers are 1, 2, 3 and 6. Thus, we can see that 6 is the highest number from these divisors. Thus, 6 is the HCF of two numbers. In case of algebraic expressions, say $6{{x}^{3}}y{{z}^{2}}$ and 3xyz, we first find the HCF of the constant parts (3 and 6). This would clearly be 3. Now, we find the terms of both algebraic expressions. Clearly, xyz is common to both the terms. Thus, HCF in this case becomes 3xyz.
Thus, we can tell that the highest common factor (HCF) is the largest number or expression which is a factor of both. Hence, the correct answer is (a).


Note: An alternative way to find HCF is by using the following property HCF $\times $ LCM (Least common multiple) = product of the numbers. LCM is the number which is the least common multiple to both the numbers (in case of 45 and 30, LCM is 90). Thus, to explain the above property, we take examples of 30 and 45. The LCM is 90, and the product of the numbers is 1350. Thus, we have –
HCF $\times $ 90 = 1350
HCF = 15
Thus, this is an alternative method to find the HCF.