Parity is the single most tested number-property idea on the GRE. An even integer is any multiple of 2 (including 0 and negatives like −6); an odd integer leaves remainder 1 on division by 2. The rules are worth memorising as reflexes: even ± even = even, odd ± odd = even, odd ± even = odd; any product touching an even factor is even, and only odd × odd stays odd. Consecutive integers alternate parity, so among any 2 consecutive integers exactly one is even, and among any n consecutive integers the product is always divisible by n! (for example, the product of 3 consecutive integers is divisible by 6). A frequent trap: "the sum of two integers is even" does not tell you they are both even — they could both be odd. Test both possibilities before you commit.
✅ Solved examples
1. If n is an integer, is n(n + 1) always even?
n and n + 1 are consecutive, so one of them is even. A product with an even factor is even. Yes — n(n + 1) is always even.
2. The sum of 5 consecutive integers is 5 times what?
Five consecutive integers are m−2, m−1, m, m+1, m+2 with middle m. Their sum is 5m, i.e. 5 times the middle (median) term. So the sum is always a multiple of 5.
3. If x is odd and y is even, what is the parity of 3x + 2y?
3x = odd × odd = odd. 2y = even. odd + even = odd. So 3x + 2y is odd.
4. Quantitative Comparison. x and y are integers with x + y odd. Quantity A: x·y. Quantity B: an even number. Which is greater / relationship?
If x + y is odd, one of x, y is even and the other odd, so their product x·y is even. Quantity A is even, matching Quantity B's parity — but the actual values are unknown, so the relationship cannot be determined.
✏️ Practice — try these, take hints as needed
1. If a is even and b is odd, what is the parity of a + b + 1?
even + odd = odd.
odd + 1 = even.
Track step by step.
Even
2. The product of three consecutive integers is always divisible by what number?
One factor is even.
One factor is a multiple of 3.
2 × 3.
6
3. If n is odd, is n² + n even or odd?
n² is odd (odd×odd).
odd + odd.
= even.
Even
4. The sum of two integers is 21 (odd). Can both be even?
even + even = even.
21 is odd.
One must be odd.
No
5. Four consecutive integers start at n. Is their sum even or odd?
Sum = 4n + 6.
4n is even, 6 is even.
even + even.
Even
📝 Topic test — 8 questions
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