Problem of the Week

Problem 19 Solution

Congratulations to Felix Reiter from Bottisham Village College for successfully solving this problem!

The key idea here is to split our problem into cases.

To solve the problem of finding two coprime integers $a$ and $b$ such that $a + b = n$ , we consider cases based on the properties of $n$ :

Case 1: $n$ is odd
Write $n = 2k + 1$ for some integer $k$ .
Choose: $a = k$ , $b = k + 1$ .
Since $a$ and $b$ are consecutive integers, they are coprime.

Case 2: $n$ is even and divisible by $4$
Write $n = 4k$ for some integer $k$ .
Choose: $a = 2k - 1$ , $b = 2k + 1$ .
These differ by $2$ , so their only possible common factor is $2$ , but both are odd, so they are coprime.

Case 3: $n$ is even but not divisible by $4$
Write $n = 2k$ , where $k$ is odd.
Choose: $a = k - 2$ , $b = k + 2$ .
These differ by $4$ , so their only possible common factors are $2$ or $4$ , but $k$ is odd, so $a$ and $b$ are both odd and coprime.

Problem of the Week

Problem 19 Solution

Site Search