Grades 5-6 Video Solutions 2015-Level 5&6 Video Solutions 2015 problem21

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Question number 21. The sum of four natural numbers is 39. The product of two of these numbers is equal to 80, and the product of the other two numbers is also equal to 80.  What is the largest of these four numbers? Since we have to factor 80 into pairs of natural numbers, let's factor 80 completely by writing down its prime factorization.  So 80 is 2 to the 4th power times 5, and then this will allow us to write down all the possible factorizations into two natural numbers.  So most trivially, we can say that's 1 times 80. We can move on to 2 times 40, 4 times 20, and then 8 times 10, and finally 16 times 5. Now, out of these five factorizations,  we have to pick two, so two pairs for a total of four numbers whose sum will be 39. Now, because 39 is odd, this is what we're looking for. 39 is odd.  We need to begin by focusing on the odd numbers as the terms in our sum because most of the numbers we see here are even, so their sum would be even, but our sum has to be odd.  And clearly, we can discount here the first choice. 1 and 80 already add up to 81. That's too much. The other possibility, the only one remaining, is 16 and 5.  We need, again, an odd number in our sum, so 16 plus 5 gives us 21,  and we need 39 minus 21 or 18 more to obtain our answer. And which of these remaining pairs actually adds up to 18? Well, that should be pretty obvious.  That is going to be 8 and 10. So 39, we have written that as 5 plus 8 plus 10 plus 16. And of these four numbers, 16 is clearly the greatest,  and that is our answer here to number 21. C is 16.

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