Integers & Number Properties • Topic 5 of 5

Properties of Zero & One

Zero and one carry special rules the GRE loves to test at the edges. Zero is an integer and is even, but it is neither positive nor negative; anything times 0 is 0; 0 divided by any nonzero number is 0; but division by 0 is undefined — a fast way the test creates "cannot be determined" traps. For exponents, any nonzero number to the 0 power is 1, and 0 to a positive power is 0 (0⁰ is not tested). One is the multiplicative identity (x·1 = x) and 1 raised to any power stays 1, while −1 alternates: (−1)^even = 1, (−1)^odd = −1. Also remember 1 is not prime. When a problem allows a denominator or a divisor to be a variable, always ask "could it be zero?" — that question alone resolves many Quantitative Comparison items.

✅ Solved examples

1. Is 0 even, odd, or neither?
0 is a multiple of 2 (0 = 2 × 0), so 0 is even. It is, however, neither positive nor negative.
2. Evaluate 7⁰ + 0⁵.
Any nonzero base to the 0 power is 1, so 7⁰ = 1. 0 to a positive power is 0, so 0⁵ = 0. Sum = 1.
3. For which value of x is the expression 5/(x − 3) undefined?
A fraction is undefined when its denominator is 0. x − 3 = 0 gives x = 3. So it is undefined at x = 3.
4. Quantitative Comparison. x is a nonzero number. Quantity A: x⁰. Quantity B: 1. Which is greater?
Any nonzero number to the 0 power equals 1, so Quantity A = 1 = Quantity B. They are equal (the "nonzero" condition is what makes this safe).

✏️ Practice — try these, take hints as needed

1. What is (−1)^50?
Even exponent.
(−1)^even.
= 1.
1
2. Evaluate 0 × 999 + 12⁰.
0 times anything is 0.
12⁰ = 1.
Add.
1
3. For which x is 8/x undefined?
Denominator zero.
x = 0.
Division by 0.
x = 0
4. Is 1 a prime number?
Primes have exactly two factors.
1 has only one factor.
So not prime.
No
5. What is (−1)^7 × 1^100?
(−1)^odd = −1.
1^100 = 1.
Multiply.
−1

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Absolute Value Take the chapter test →