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Question 2. Use your solution from Question 1 to find integers x and y such that

by admin|Published May 18, 2026

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Question 2. Use your solution from Question 1 to find integers $x$ and $y$ such that

$875x+4075y=50$

Theorem 4. If $(a,b)=d,$ then there are integers $x$ and $y$ such that

$ax+by=d$

From Q1

$4075=4(875)+575$

$875=(575)(1)+300$

$575=(300)(1)+275$

$300=275(1)+25$

$300=25(12)+0$

It is suggested to work “The Euclidean Algorithm” backwards.

Then,

$300=25(12)+0$

$\Longrightarrow 300=275(1)+25$

$\Longrightarrow 300-275(1)=25$

$\Longrightarrow [875-(575)(1)]-[575-(300)(1)]=25$

$\Longrightarrow [875-(575)(1)]-[(4075-(4)(875))-(300)(1)]=25$

Note: Get all terms to be original terms $875,4075$

$\Longrightarrow [875-(575)(1)]-[(4075-(4)(875))-(875-(575)(1))]=25$

$\Longrightarrow 875-(575)(1)-(4075-(4)(875))+(875-(575)(1))=25$

$\Longrightarrow 875-(575)(1)-4075+(4)(875)+875-(575)(1)=25$

$\Longrightarrow -4075+(6)(875)-(575)(2)=25$

$\Longrightarrow -4075+(6)(875)-(4075-(4)(875))(2)=25$

$\Longrightarrow -4075+(6)(875)-4075(2)+(8)(875)=25$

$\Longrightarrow (14)(875)+4075(-3)=25$

$\Longrightarrow (875)(28)+4075(-6)=50$

$\therefore x=28,\quad y=-6$

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