## Notes on Divisibility – Relative Primes

Recently, I asked a ninth grader if he could simplify the fraction $\frac{223}{225}$.  I was really interested in whether he could find a common divisor of 223 and 225, but asking him to simplify the fraction was a quicker way of asking the question in terms he’d seen before.  He said: “no, you can’t simplify because the two numbers are only two apart.”  I asked him why that was relevant.  He said: “Because they are only two apart, a common divisor would have to be even, and neither 223 nor 225 are even.”  Though I didn’t probe deeper, I felt confident that he wasn’t trying to find divisors for 223 and then independently for 225 and then looking for common divisors.  It wasn’t that he gave up after trying just some small number of possible divisors, no did he attempt to find the prime factorization of the numerator and the denominator.  He seemed to have invented, on the spot, the key idea underlying Euclid’s algorithm – and over the course of the rest of the hour, he worked with me on an algorithm, coming up with a description of an algorithm very similar to the one I sketched in a previous post.

I had earlier encountered students who could quickly see that e.g. $\frac{131}{132}$ could not be further simplified, but none that could articulate why or who could offer generalizations from it.

From the specific example $\frac{131}{132}$ we could generalize that $\frac{n}{n+1}$ will not simplify for any $n$, and from the example $\frac{223}{225}$ we could generalize that $\frac{2n-1}{2n+1}$ can not be further simplified.

Pairs of numbers such as $131$ and $132$ that correspond to fraction that will not simplify, such pairs of numbers are called relative prime.  The pair $223$ and $225$ are also relative prime, and so are $2n-1$ and $2n+1$.

Can you think of other generalizations for pairs of numbers who are relative primes?

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