Question

If the HCF of 408 and 1032 is expressible in the form \[1032m - 408 \times 5\], find \[m\].

Answer 1

Hint: Here we will use the concept of HCF i.e. highest common factor to solve the question. First, we will find the HCF of the given two numbers using the Euclid's division algorithm. Then we will equate the HCF to the given form of the HCF. Further, we will solve the obtained equation to get the value of \[m\].

Complete step by step solution:
Given two numbers are 408 and 1032.
Firstly we will find the HCF of the given two numbers.
Therefore, by using the Euclid's division algorithm we get
\[\begin{array}{l}1032 = 408 \times 2 + 216\\408 = 216 \times 1 + 192\\216 = 192 \times 1 + 24\\192 = 24 \times 8 + 0\end{array}\]
So, we know when the remainder becomes zero then the divisor is the HCF of the numbers. Therefore, the HCF of the numbers 408 and 1032 is 24.
It is given that the HCF is in the form of \[1032m - 408 \times 5\]. Now we will equate this form to 24 to get the value of \[m\]. Therefore, we get
\[ \Rightarrow 1032m - 408 \times 5 = 24\]
\[ \Rightarrow 1032m - 2040 = 24\]
\[ \Rightarrow 1032m = 24 + 2040 = 2064\]
Now, by solving the above equation we get
\[ \Rightarrow m = \dfrac{{2064}}{{1032}} = 2\]
Hence, the value of \[m\] is 2.

Note: Here we should note that HCF is the short form of highest common factor which is the largest factor which is the largest common divisor of both the numbers. We should calculate the HCF of the numbers carefully. We can also calculate the HCF by taking all the common factors of the given numbers but by Euclid's division algorithm we will be able to find the HCF more quickly than the other conventional method. Also we should know that LCM is the lowest common factor which is exactly divisible by both the numbers.