prove that number integer x,y exit satisfying x+y=200 and (x,y) = 7
Answers
Lets split the numbers into factors and see where we can see a fit for the answer choices.
A): As you can see, 5 is already the GCF of 20 and 35. So 20y and 35x have at least 5 as GCF.
Incorrect.
B): As discussed in A), 5 is already part of the GCF. We only have to look whether we can bring in (x-y) into both numbers. And yes, indeed, thats possible. If x=2y, then (x-y)=(2y-y)=y is included in both, x and y!
Incorrect.
C): Well, we would need 20 and x as factors in both of them(35x and 20y). 35x lacks the 20, so x has to bring two 2s. But then, for x also to be included in the GCF, y would have to include x (which is possible) and x would have to include itself, in addition to two 2s, because x also has to bring in those two 2s.. So paradoxically, x needs to be at least 20x, which is not possible.
Correct.
D): 20y requires two 2s and one 5 and one y. If all of them would be included in 35x, we would be able to call at least 20y the GCF. So lets make 35x include 20y. All we need for that is two 2s and y. If we include at least them in x, then at least 20y is the GCF.
Incorrect.
E): We can play the same game as in D. For 35x to be the GCF, let 35x be included in 20y. Therefore, y needs to have at least 7x.
Incorrect.