If $a$ and $b$ are coprime integers, which of the following pairs of integers must also be coprime?

Step 1 (Understand Coprimality): Two integers $a$ and $b$ are coprime if and only if their greatest common divisor is 1, i.e., $\gcd (a,b)=1$.

Step 2 (Apply GCD properties): By the Euclidean algorithm / subtraction properties of GCD:

Since we are given that $\gcd (a,b)=1$, we must have:

This mathematically guarantees that the pair $a$ and $a+b$ are coprime.

Step 3 (Analyze other options):

• For $a$ and $ab$: If $a>1$, then $\gcd (a,ab)=a\ne 1$, so they are not coprime.

• For $a+b$ and $a-b$: If $a=3,b=1$ (coprime), then $a+b=4$ and $a-b=2$, which have $\gcd (4,2)=2\ne 1$ (not coprime).

Step 4 (Conclusion): The correct pair is a and a+b, which corresponds to option index 0.