SOLUTION: Three friends are packing sweets into gift boxes. They agree that each box should contain the same number of sweets, but they are each working in separate locations with their own

Question 1182881: Three friends are packing sweets into gift boxes. They agree that each box should contain the same number of sweets, but they are each working in separate locations with their own pile of sweets so cannot share boxes.
Gwen has 286 sweets, Bill has 390 sweets and Zeta has 468 sweets. If they put the largest number of sweets into each box that they can and they use up all their sweets, how many boxes of sweets will they pack?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52786) (Show Source): You can put this solution on YOUR website!
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Three friends are packing sweets into gift boxes. They agree that each box should contain the same number of sweets,
but they are each working in separate locations with their own pile of sweets so cannot share boxes.
Gwen has 286 sweets, Bill has 390 sweets and Zeta has 468 sweets. If they put the largest number of sweets
into each box that they can and they use up all their sweets, how many boxes of sweets will they pack?
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This problem is to find the Greatest Common Divisor (GCD) of the numbers 286, 390 and 468. These numbers have common divisors 13 and 2 and have no other prime common divisors. So, their greatest common divisor is 13*2 = 26. THEREFORE, the three friends should put 26 sweets into each box. Doing this way, they will fill = 44 boxes. ANSWER

Solved.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The basic problem here is to find the greatest common factor of 286, 390, and 468. The response from the other tutor just tells you it is 26. Thinking you might need help finding the GCF, here is a discussion of some ways to do it.

In a math class, you will probably be told to find the prime factorization of each number and identify all the factors common to all three. That is not the fastest path for most students; I would modify the process.

Instead of starting with each number and finding its prime factorization, make the process easier and faster by finding the prime factorization of one of the numbers and then look for those prime factors in the other numbers.

I would start with the 390, because the final 0 means finding the prime factorization will be fast.

390 = 39*10 = (3*13)*(2*5) = 2*3*5*13

Now, instead of starting from scratch on the other two numbers, look for those same prime factors in each of them. 286 and 468 clearly both have prime factors of 2; and clearly neither has a prime factor of 5. Using the rule for divisibility by 3 shows that they don't both have a prime factor of 3, so the only prime factor left is 13. Calculations then show that both 286 and 468 have a prime factor of 13.

So the GCF of the three numbers is 2*13=26.

Here is another way to find the GCF of the three numbers.

If all three numbers have a common factor, then the difference between any two of the numbers will have that common factor. We can use that fact, sometimes repeatedly, to find the GCF.

For these three numbers, the process might go like this:

390-286=104
468-390=78
104-78=26

Then once you have found the GCF, finish the problem as shown in the response from the other tutor.

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