Question

Express the H.C.F. of 48 and 18 as a linear combination.

Answer 1

Hint: Proceed the solution of this question first writing the given numbers in Euclid’s division algorithm form then keep on factorising the divisor till we got remainder zero and the required HCF. Then writing HCF with the help of previous factors we can write in the form of linear combination.

Complete step-by-step answer:
In the question, it is asked the HCF of 48 and 18 and we have to express the HCF of 48 and 18 as a linear combination
We know that Euclid’s division algorithm is the process of applying Euclid’s division lemma in succession several times to obtain the HCF of any two numbers.
To understand this algorithm, suppose that there are two numbers: a and b. applying Euclid’s division lemma, we will have two integers q and r such that,
a=b(q)+r, where, q is the quotient and r is the remainder
Hence we have a=48 and b=18
Hence we can write
⇒48=18×2+12
⇒18=12×1+6 …………. (1)
⇒12=6×2+0
∴HCF (18,48) =6

Now,
 we can write value of 6 from above expression (1)
⇒6=18−12×1
⇒6=18−(48−18×2)
⇒6=18−48×1+18×2
⇒6=18×3−48×1
⇒6=18×3+48× (−1)
⇒6=18x+48y which is one of the required linear combination expressions.
whereas, x=3, y=−1

therefore,
6=18×3+48× (−1)
6=18×3+48× (−1) +18×48−18×48
6=18(3+48) +48(−1−18)
6=18×51+48× (−19)
⇒6=18x+48y which is one of the required linear combination expressions.
whereas, x=51, y−19
Thus x and y are not unique.

Note- In this particular question, we can also verify the HCF by prime factorisation method. Which can be written as-
⇒48=2×2×2×2×3
⇒18=2×3×3
We know that the largest or greatest factor common to any two or more given natural numbers is termed as HCF of given numbers. It is also known as GCD (Greatest Common Divisor).
Hence HCF of 48 and 18 will be their multiplication of common factors which is as 2×3= 6