Question

D is the HCF of 1155 and 506. Find x and y satisfying,\[D = 1155x + 506y\], also show that x and y are not unique.

Answer 1

Hint: Here the given question needs to find the highest common factor and then solve with the equation, here to find the highest common factor we would use factorization method, and then on solving further we can solve the answer.
Formulae Used: To find the highest common factor, here we need to solve by using factorization method.

Complete step-by-step solution:
Here we need to first find the highest common factor and then solve further, on solving we get:
\[
   \Rightarrow 1155 = 506 \times 2 + 143 \\
   \Rightarrow 506 = 143 \times 3 + 77 \\
   \Rightarrow 143 = 77 \times 1 + 66 \\
   \Rightarrow 77 = 66 \times 1 + 11 \\
   \Rightarrow 66 = 11 \times 6 + 0 \]
Hence the highest common factor is eleven.
Now solving, we can write eleven in the given form, we get:
\[
   \Rightarrow 11 = 77 - 66 \times 1 \\
   \Rightarrow 11 = 77 - [143 - 77] \\
   \Rightarrow 11 = 77 - 143 + 77 \\
   \Rightarrow 11 = 77 \times 2 - [1155 - 506 \times 2] \\
   \Rightarrow 11 = [506 - 143 \times 3](2) - [1155 - 506 \times 2] \\
   \Rightarrow 11 = [506 - (1155 - 506 \times 2) \times 3](2) - [1155 - 506 \times 2] \\
   \Rightarrow 11 = 506(2) - [1155 - 506(2)](6) - 1155 + 506(2) \\
   \Rightarrow 11 = 506(2) - 1155(6) + 506(12) - 1155(1) + 506(2) \\
   \Rightarrow 11 = 506(16) - 1155(7) \\
   \Rightarrow 11 = 1155( - 7) + 506(16) \]
Now comparing it with the main equation we get:
\[ \Rightarrow D = 1155x + 506y\,and\,11 = 1155( - 7) + 506(16)\,and\,D = 11\]
Hence,
\[ \Rightarrow x = - 7,\,y = 16\]
It is our required answer, and here we get that the value of both the variables are not the same.

Note: Here we solve the given question by using factorization method, and then splitting the numbers, here we split the value of “D” according to the equation we are provided with, and as we get the equation similar to our equation we get the answer for the variable.