1. ## Proof- GCD

A man goes to a stream with a 9–pint container and a 16–pint
container. What should he do to get 1 pint of water in the 16–pint container?

2. Originally Posted by anncar
A man goes to a stream with a 9–pint container and a 16–pint
container. What should he do to get 1 pint of water in the 16–pint container?
It seems you need to solve the equation,
$9x+16y=1$ for integers $x,y$ the positives will represent that you are adding and negatives that you are removing.
One such solution is $x=16,y=-7$

3. Originally Posted by anncar
A man goes to a stream with a 9–pint container and a 16–pint
container. What should he do to get 1 pint of water in the 16–pint container?
this was quite difficult, there may be another way, i'm notorious for doing things the hard way:

Step 1: fill the 16 pt and pour it into the 9 pt (you now have 9 pts in the 9 pt container and 7 pts in the 16 pt container)

Step 2: empty the 9 pt and pour the 7 pts from the 16 pt container into it. (you now have 7 pts in the 9 pt container and 0 pts in the 16 pt container)

Step 3: fill the 16 pt and fill the 9 pt (you now have 9 pts in the 9 pt container and 14 pts in the 16 pt container)

Step 4: empty the 9 pt container, and again, fill it with what remains in the 16 pt container (you now have 9 pts in the 9 pt container and 5 pts in the 16 pt container)

Step 5: empty the 9 pt container and pour the water remaining in the 16 pt container into it (you now have 5 pts in the 9 pt container and 0 pts in the 16 pt container)

Step 6: fill the 16 pt and then fill the 9 pt container with that water (you now have 9 pts in the 9 pt container and 12 pts in the 16 pt container)

Step 7: empty the 9 pt container and again fill it with what remains in the 16 pt container (you now have 9 pts in the 9 pt container and 3 pts in the 16 pt container)

Step 8: empty the 9 pt container and empty the 16 pt container into the 9 pt container (you now have 3 pts in the 9 pt container and 0 pts in the 16 pt container)

Step 9: fill the 16 pt and pour as much as possible into the 9 pt (you now have 9 pts in the 9 pt container and 10 pts in the 16 pt container)

Step 10: finally, empty the 9 pt container and then pour as much as possible out of the 16 pt container into the 9 pt container (you now have 9 pts in the 9 pt container and 1 pts in the 16 pt container)

that's it in 10 steps. there may be a way to do it in fewer