An integer is odd when it cannot be split into two equal integer groups. In decimal notation, its last digit is 1, 3, 5, 7, or 9.
Start with the last digit
For an integer written in ordinary decimal notation, look at the final digit. A final 1, 3, 5, 7, or 9 means odd. A final 0, 2, 4, 6, or 8 means even. This is the familiar divisibility test for two described in the College of the Desert’s number theory material.
For example, 381 is odd because it ends in 1. The earlier digits do not need to be added, multiplied, or interviewed. The number 6,284 is even because it ends in 4. A minus sign does not change the answer: −381 is odd too.
Check by making pairs
Imagine 13 counters. Arrange them into pairs, and you get six complete pairs with one counter left. That leftover counter identifies an odd count. With 14 counters, all seven pairs are complete.
This picture is useful for positive counts. For negative integers, use the arithmetic definition instead of trying to assemble a negative pile of counters.
Another check is division: 381 ÷ 2 = 190.5, so the result is not an integer. By comparison, 382 ÷ 2 = 191, which is an integer. Apply this test only after establishing that the original value is an integer.
Write down the proof
An odd integer has the form n = 2k + 1, where k is an integer. For 381, choose k = 190:
381 = 2 × 190 + 1.
For −27, choose k = −14:
−27 = 2 × (−14) + 1.
These equations are exact demonstrations, not estimates. Finding one suitable integer value of k is enough.
Check the input before the answer
Parity applies to integers. The number 3.7 is neither odd nor even under the usual definition. The notation 3.0, however, represents the integer 3, so that value is odd. A decimal point alone does not decide anything.
The calculator accepts decimal integer notation without a decimal point. Enter 3 for the value written as 3.0, and enter 6284 without a thousands separator. See decimals and parity for the distinction between a number and its spelling.
Try 205, −48, and 0 before checking: the answers are odd, even, and even. Zero passes the same rule as every other integer; it does not need its own exemption form.