# A Guide to the GRE/Odds and Evens

## Odds & Evens

[edit | edit source]### Adding & Subtracting

[edit | edit source]**An odd plus an odd equals an even.**

5 + 7 = 12 = even

**An even plus an even equals an even.**

6 + 4 = 10 = even

**An odd plus an even equals an odd.**

6 + 5 = 11 = odd

### Multiplying & Dividing

[edit | edit source]**An odd multiplied by an odd equals an odd.**

5(5) = 25 = odd

**An odd multiplied by an even equals an even.**

5(2) = 10 = even

**An even multiplied by an even equals an even.**

4(8) = 32 = even

**When in doubt, test this with two known numbers. If unsure, try 1 + 3 or (2)(5) and see what type of number it yields.**

### Practice

[edit | edit source]1. *xy* is even and *xz* is odd. Is *y* odd or even?

2. *a* + *b* + *c* is even. If *a* is odd, then could *b* and *c* be

both odd?

both even?

odd and even?

3. (*f* + *g*)*h* is an odd number. If *f* is even, is *g* + *h* odd or even?

### Answers to Practice Questions

[edit | edit source]1. *y* must be even.

The product of two numbers is only odd if both numbers are odd. Since *xz* is odd, both *x* and *z* must be odd. However, *xy* is even. This means that between *x* and *y*, one must be even. Since *x* is odd, *y* must be even.

2. *b* and *c* must be odd and even, not necessarily in that order.

An odd plus an odd equals an even, as does an even plus an even. Only an odd plus an even equals an odd. Since *a* + *b* + *c* is even and *a* is odd, *b* + *c* must be odd. This means that, between *b* and *c*, one must be odd, and one must be even. If they were both odd or even, *b* + *c* would be even, meaning *a* + *b* + *c* would be even, but it isn't.

3. *g* + *h* must be even.

Only an odd and another odd multiply to make an odd. Therefore, since (*f* + *g*)*h* is odd, *f* + *g* must be odd. Since *f* is even, *g* must be odd, since only an even and an odd add up to an odd. *h* also must be odd, since it multiplies to make an odd. Therefore, since an odd plus an odd equals an even, *g* + *h* must be even.