SOLUTION: Joe and Anna collect football cards. The GCF of the numbers of cards in their collections is 15. Altogether Joe and Anna have 75 Cards. If Joe has more cards than Anna how many car

Question 207188: Joe and Anna collect football cards. The GCF of the numbers of cards in their collections is 15. Altogether Joe and Anna have 75 Cards. If Joe has more cards than Anna how many cards do they each have
Answer by mickclns(59) (Show Source): You can put this solution on YOUR website!
Let J and A be the numbers of cards that Joe and Anna have, respectively. Let J'=J/15 and A'=A/15. Since 15 is the GCF of J and A, the GCF of J' and A' is 1 (why?).
Then J+A = 75 and J'+A' = J/15 + A/15 = (J+A)/15 = 75/15 = 5. J'>A' and J' and A' are positive integers (why?). There are two possible solutions: either J' is 4 and A' is 1, in which case J is 60 and A is 15 (why?); or J' is 3 and A' is 2, in which case J is 45 and A is 30.
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Another way to do this is to list the multiples of 15 up to 75 (J and A must both be multiples of 15 and if a number is not a multiple of 15 then either it is not J or not A, or is neither -- why?).
15, 30, 45, 60, 75
Find a pair in this list that add to 75. There are two such pairs, (60 and 15) and (45 and 30).
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I have shown two alternative methods for finding J and A. The first is more appropriate for someone who has been introduced to GCF before and can't quite bring the knowledge together to solve it. The second is more appropriate for someone who is just learning about GCF for the first time.
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