Explanation

If the variables are positive, Quantity A is greater. However, all three variables could equal zero, in which case the two quantities are equal. Watch out for the word “non-negative,” which means “positive or zero.”

Let's test some possible values

case i: If k = 2m = 4n, then it COULD be the case that k = m = n = 0
We get:
Quantity A: km = (0)(0) = 0
Quantity B: kn = (0)(0) = 0
In this case, the two quantities are equal


case ii: If k = 2m = 4n, then it COULD be the case that k = 4, m = 2 and n = 1
We get:
Quantity A: km = (4)(2) = 8
Quantity B: kn = (4)(1) = 4
In this case, Quantity A is greater

Answer: D

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Let's test some possible values

case i: If k = 2m = 4n, then it COULD be the case that k = m = n = 0
We get:
Quantity A: km = (0)(0) = 0
Quantity B: kn = (0)(0) = 0
In this case, the two quantities are equal


case ii: If k = 2m = 4n, then it COULD be the case that k = 4, m = 2 and n = 1
We get:
Quantity A: km = (4)(2) = 8
Quantity B: kn = (4)(1) = 4
In this case, Quantity A is greater

Answer: D

case i: If k = 2m = 4n, then it COULD be the case that k = m = n = 0 ------How "0"?

I'm not sure what you're asking.
Can you elaborate?

sorry, my mistake. it will be
case i: If k = 2m = 4n, then it COULD be the case that k = m = n = 0 ------How "0", means how they could be '0' Where as it is said that k = 2m = 4n?

If k = 0, m = 0, and n = 0, then we get: 0 = 2(0) = 4(0), which is true.
In other words, k = 0, m = 0, and n = 0 is a solution to the equation k = 2m = 4n

Does that help?

They got me with this one. Just like with 2 being the smallest prime. Ugh.

What will be the answers if non negative integer is replaced by positive numbers in the questions..

Let's find out.
Here is the NEW question:

k = 2m = 4n, and k, m, and n are POSITIVE integers.
Given:
QUANTITY A: km
QUANTITY B: kn

Since we know that k is positive, we can safely divide both sides by k to get:
QUANTITY A: m
QUANTITY B: n

Next, since we're told 2m = 4n, we can divide both sides of the equation by 2 to get: m = 2n
So, for Quantity A, we can replace m with 2n to get:
QUANTITY A: 2n
QUANTITY B: n

From here, we might already see that Quantity A is greater. However we can also confirm this by subtracting n from both quantities to get:
QUANTITY A: n
QUANTITY B: 0

Since n is now positive (for this particular question), we can see that Quantity A is greater.

Answer: A

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