8. What is the smallest number of gallons you can measure with a 72 gal bucket and a 21 gal bucket, and how would you measure this amount?

The smallest number of gallons will be the GCF(72, 21). We can use the Euclidean Algorithm to find it

The last nonzero remainder is 3, so the samllest number of gallons that can be measured out is 3.

We can use the Euclidean Algorithm to find out how to measure it.

3 = 21 - 2x9

However, from the problem before we see that

9 = 72 - 3x21

Substitute this in for 9 in the previous equation.

3 = 21 - 2(72 - 3x21)

Remove parentheses

3 = 21 - 2x72 + 6x21

Combine like terms.

3 = 7x21 - 2x72

= 147 - 144

You would dump 7 of the 21 gallon buckets into the 72 gallon bucket; each time the 72 gallon bucket gets full, dump it out and start filling it up again. After you have filled the 72 gallon bucket for the 2nd time, there will be 3 gallons left in the 7th 21 gallon bucket.