Hello! Can someone help me out with this problem from the 5 lb. Manhattan book.

The answer key says the answer is D. But I don’t understand how the variable can be 0 and still satisfy the equation. For example if k=0 then wouldn’t k=m which is also 0.

I mean the coefficients wouldn’t really be playing a role there would they.

I might be wrong, so if someone can please explain it would be highly appreciated.

I think you’re asking if k,m and n can be the same integer (i.e. 0), right?

Im not a 100% sure but I think ETS would typically clarify if they are all “different integers” but since it doesn’t say so here I think we can take the values of k (and subsequently m and n) to be 0 as one situation, and take m=4 and n=2 as second situation.

Hence D.

Actually k is not equal to m its still equal to 2m but if m = 0 k = 2* 0 thus the value of 2m is 0 which means k is also zero. But the relation still satisfies

You are right: when k=0, then k=2m=4n implies that m=0 and n=0. Thus, km=0=kn.

But the case you mentioned is merely one example. Since k, m, n can be *any* nonnegative integers, we must consider other possible cases. For example:

When k=4, then k=2m=4n implies that m=2 and n=1. Thus, km=4\cdot2=8 > 4\cdot 1 = kn.

We can have km=kn or km>kn depending on the values chosen for k. So, the answer is (D).