Answer to Question #266114 in Combinatorics | Number Theory for crane 67

Question

For each pair of numbers find integers 𝑥 and 𝑦 such that 𝑎𝑥 + 𝑏𝑦 =gcd (𝑎,𝑏)

a) 𝑎= 91, 𝑏=10

Answer 1

For the pair $𝑎= 91, 𝑏=10$ of numbers let us find integers $x$ and $y$ such that $𝑎𝑥 + 𝑏𝑦 =gcd (𝑎,𝑏).$

Taking into account that $91=10\cdot9+1$ and $10=10\cdot 1+0,$ we conclude that $gcd (91,10)=1=91-10\cdot 9=91\cdot 1+10\cdot(-9).$ Consequently, $x=1,\ y=-9.$