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Number Theory

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Moderate

Bezout's Identity

NUMTH-KE7AYK

You may use Bezout's identity:

If $a$ and $b$ are integers with the greatest common divisor $d=(a,b)$, then there exist integers $x$ and $y$ such that

$$ax+by=d$$

Also, we say that $(x,y)$ is an integer solution to an equation $f(x,y)=0$ if $x$, $y$ satisfies $f(x,y)=0$ and $x$, $y$ are integers. Suppose that $(x,y)\in B$. We say that $f(x,y)$ represents the elements in a set $A$ if for any $a\in A$, there is $(x,y)\in B$ such that $f(x,y)=a$.

Which of the following statements are true?

Check ALL that apply.

A

If $x$ and $y$ are positive integers, then $2x+3y$ represents all positive integers.

B

If $x$ and $y$ are coprime integers, then $4x+6y$ represents all even integers.

C

The equation $3x+4y=5$ has infinitely many integer solutions.

D

The equation $12x+18y=4$ has infinitely many integer solutions.

E

The equation $12x+18y=6$ has finitely many integer solutions.