How does Euclid's algorithm give solutions to equations?
Euclid's algorithm is really useful to be able to, firstly, see if two numbers are co-prime, in other words to see if they share any common factors, but also to find solutions to equations. Say we have two integers that satisfy: 32x + 24y = 16 Then we use Euclid's algorithm to first calculate the greatest common divisor (gcd) of 32 and 24. Hopefully, the method of this is ok? So we get gcd(32,24) = 8. Now, we can reverse what we did to get our solutions to the equation above. But don't forget that we had the equation equal to 16, not 8. This is often used in exams to trip up students, so look out for that.
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