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An Introduction to the Theory of Numbers

Exercise 1.2.25 (There are infinitely many pairs of integers $x,y$ satisfying $x + y = 100$ and $(x,y) = 5$)

Prove that there are infinitely many pairs of integers x , y satisfying x + y = 100 and ( x , y ) = 5 .

Answers

Notations : Here x ∧ y = gcd ( x , y ) and ( x , y ) is an ordered pair.

Proof. If x ∧ y = 5 , then x = 5 X , y = 5 Y for some integers X , Y such that X ∧ Y = 1 . Then, under the hypothesis x ∧ y = 5 ,

x + y = 100 ⟺ 5 X + 5 Y = 100 ⟺ X + Y = 20 .

( X , Y ) = ( 9 , 11 ) is a particular solution satisfying X ∧ Y = 1 , thus ( x , y ) = ( 45 , 55 ) is a solution of x ∧ y = 5 , x + y = 100 .

For any integer k ∈ ℤ , the pair ( x , y ) = ( 45 + 100 k , 55 − 100 k ) satisfy x + y = 100 . Moreover, using the rule a ∧ b = a ∧ ( a + b ) = a ∧ ( a + kb ) ,

x ∧ y = ( 45 + 100 k ) ∧ ( 55 − 100 k ) = ( 45 + 100 k ) ∧ [ ( 45 + 100 k ) + ( 55 − 100 k ) ] = ( 45 + 100 k ) ∧ 100 = 45 ∧ 100 = 5

We obtain infinitely many solutions of x + y = 100 , x ∧ y = 5 , of the form

x = 45 + 100 k , y = 55 − 100 k , k ∈ ℤ .

(There are many others solutions.) □

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2024-09-28 09:58
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