The GRE tests your knowledge of how odd and even numbers combine through addition, subtraction, multiplication, and division. Rules for adding, subtracting, multiplying, and dividing odd and even numbers can be derived by simply picking numbers and testing them out. While this is certainly a valid strategy, it also pays to memorize the following rules for operating with odds and evens, as they are extremely useful for certain GRE math questions.
Add or subtract 2 odds or 2 evens, and the result is EVEN: 7 + 11 = 18 and 14 − 6 = 8.
Add or subtract an odd with an even, and the result is ODD: 7 + 8 = 15 and 12 − 5 = 7.
When you multiply integers, if any of the integers are even, the result is even: 3 × 8 × 9 × 13 = 2,808.
Likewise, if none of the integers are even, then the result is odd: 3 × 5 × 7 = 105.
If you multiply together several even integers, the result will be divisible by higher and higher powers of 2. This is because each even number will contribute at least one 2 to the factors of the product.
For example, if there are two even integers in a set of integers being multiplied together, the result will be divisible by 4: 2 × 5 × 6 = 60 (divisible by 4).
If there are THREE even integers in a set of integers being multiplied together, the result will be divisible by 8: 2 × 5 × 6 × 10 = 600 (divisible by 8).
To summarize so far:
| Odd ± Even = Odd | Odd × Odd = Odd |
| Odd ± Odd = Even | Even × Even = Even (and divisible by 4) |
| Even ± Even = Even | Odd × Even = Even |
There are no guaranteed outcomes in division, because the division of two integers may not yield an integer result. There are several potential outcomes, depending upon the value of the dividend and divisor.
Divisibility of Odds & Evens
| Even? | Odd? | Non-Integer? | |
| Even ÷ Even | ✓ Example: 12 ÷ 2 = 6 |
✓ Example: 12 ÷ 4 = 3 |
✓ Example: 12 ÷ 8 = 1.5 |
| Even ÷ Odd | ✓ Example: 12 ÷ 3 = 4 |
✗ | ✓ Example: 12 ÷ 5 = 2.4 |
| Odd ÷ Even | ✗ | ✗ | ✓ Example: 9 ÷ 6 = 1.5 |
| Odd ÷ Odd | ✗ | ✓ Example: 15 ÷ 5 = 3 |
✓ Example: 15 ÷ 25 = 0.6 |
An odd number divided by any other integer cannot produce an even integer. Also, an odd number divided by an even number cannot produce an integer, because the odd number will never be divisible by the factor of 2 concealed within the even number.
For questions #1–3, say whether the expression will be odd or even.
1,007,425 × 305,313 + 2
5 × 778 × 3 × 4 + 1
The sum of four consecutive integers.
Will the product of two odd integers divided by a multiple of two be an integer?