PSAT/SAT & ACT Math Attack – Number Properties

Pythagoras

The famous mathematician Pythagoras, as depicted in Raphael’s “The School of Athens.”

“If xyz =  0 and x < y < z, which of the following CANNOT be true?”

A) x^2 = z^2

B) xy = yz

C) xy > 0

D) xz > 0

E) yz > 0

When multiplied, three numbers, x, y, and z, give a product of zero. This means one of them (and only one of them, since none of them are equal to one another) must be zero, since the only way to get zero as a product of multiplication is to multiply by zero. Taking the given inequality into account, we can deduce three things:

If x is zero, then y and z must be positive numbers.

If y is zero, then x must be a negative number and z must be a positive number.

If z is zero, then x and y must be negative numbers.

The easiest way to see if any of the answer choices could be true is to think of examples using real numbers that would make them true. If we come across one where we can’t think of an example, then that one must be the answer. Let’s look at choice (A):

A) x^2 = z^2

Under what circumstance could two different numbers be equal to each other when squared? Well, we know that the square of a negative number is the same as the square of the positive number with the same absolute value: (-2)^2 is the same as 2^2; they both equal 4. Could x and z have the same absolute value? Well, if we let y be zero (which we said was a possibility), then x could be -2 and z could be 2. Thus, (A) is possible, so it is not the correct answer to this problem. Next, consider answer choice (B):

Numbers!

Numbers!

B) xy = yz

If y equals zero, then both xy and yz would also have to be zero, since any number multiplied by zero is zero. Thus, (B) could also be possible. Next, let’s examine choice (C):

C) xy > 0

Could this be true? In order for the product of x and y to be greater than zero, neither x nor y could equal zero, which means that z would have to be zero in this scenario, and x and y would both have to be negative. A negative number multiplied by another negative number always yields a positive result, and positive numbers are greater than zero, so (C) is definitely possible. Now, consider answer choice (D):

D) xz > 0

Could the product of x and z be greater than zero? In order for that to be true, neither x nor z could equal zero, which would mean that y would have to be zero and x would have to be negative and z would have to be positive. A negative number multiplied by a positive number always yields a negative result, which is less than zero, so it does not seem like (D) is possible. This looks like our answer, but just to be sure, let’s look at (E):

E) yz > 0

Once again, in order for this statement to be true, neither y nor z could equal zero, which means that x would have to equal zero, which would mean that both y and z would have to be positive. Two positive numbers always produce a positive product when multiplied by each other, so choice (E) is possible. Thus, (D) is definitely our answer.

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