The nonprofit organization at which you volunteer has receiv

The non-profit organization at which you volunteer has received a donation of 528 energy bars and 336 apples. You want to create packages to distribute such that each package contains the same number of energy bars and each package contains the same number of apples. What is the maximum number of packages you can make? Find your answer using the Euclidean algorithm (show all your steps) and explain why it is the solution. Give a complete English sentence as your answer saying how many packages you can make and each package contains how many energy bars and how many apples.

Solution

Given that there was a donation of 528 energy bars and 336 apples.

Need to create packages to distribute such that each package contains the same number of enrgy bars and each package contains same number of apples.

Find the maximum number of packages I can make.

Maximum number of packages can be find by finding the Greatest common divisor.

GCD of the numbers will be the solution because some number of packages have equal number of enegy bars and equal number of apples.That number of packages will be the GCD of the two numbers.

Given that find the answer using Euclidean algorithm.

Euclidean Algorithm:

Given 528 energy bars and 336 apples.

multiples of 336 are subtracted from 528 until the remainder is less than 336. One such multiples can be subtracted, leaving a remainder of 192:

528 = 1 × 336 + 192.

Then multiples of 192 are subtracted from 336 until the remainder is less than 192. One such multiple can be subtracted, leaving a remainder of 144:

336 = 1 × 192 + 144.

Then multiples of 144 are subtracted from 192 until the remainder is less than 144. One such multiple can be subtracted, leaving remainder 48:

192 = 1 × 144 + 48.

Then multiples of 48 are subtracted from 144 until the remainder is less than 48. Three such multiples can be subtracted, leaving no remainder :

144 = 3 × 48 + 0.

Since the last remainder is zero, the algorithm ends with 48 as the greatest common divisor of 528 and 336.

Therefore, GCD(528, 336) = 48

So, number of packages I can make is 48.

Number of energy bars in each package is 528/48 = 11

Number of apples in each package is 336/48 = 7

Therefore, Total of 48 packages I can make and each package contains 11 energy bars and 7 apples.