A nonnegative integer in binary is odd when its final bit is 1. Every other binary place is worth an even power-of-two amount.
Look at the rightmost bit
In binary notation, a nonnegative integer is odd when its final digit is 1 and even when its final digit is 0. For example, binary 101101 is odd, while binary 101100 is even.
Binary uses powers of two for its place values: from right to left, they are 1, 2, 4, 8, 16, 32, and so on. This positional system is explained in Stephen Davies’s binary-number chapter.
Expand one example
The binary numeral 101101 represents:
32 + 8 + 4 + 1 = 45.
The contributions 32, 8, and 4 are all even. Their sum is even, and the final 1 makes the total odd. Removing that final contribution gives 101100, which represents 44.
This works for every nonnegative binary integer. All positions except the rightmost have weights divisible by two. Their total can be written as 2k. The final bit adds either zero or one, producing 2k or 2k + 1.
A long string of bits does not require a long proof. The argument already covers every earlier position.
Value parity is different from bit-count parity
The phrase parity bit sometimes refers to whether a bit string contains an odd or even number of 1s. That is a different question from whether the represented integer is odd.
For example, binary 11 contains two 1s, so its count of set bits is even. But its numerical value is 3, which is odd. Binary 100 contains one 1, yet represents the even integer 4.
To classify the integer’s value, inspect the rightmost bit. To classify the number of set bits, count them. These tasks share terminology, which is thoughtful of nobody.
Negative numbers and code
A minus sign applied to a binary magnitude does not change parity: −101 in signed mathematical binary notation represents −5, an odd integer. When working with a machine’s signed representation, specify the representation before interpreting its bit pattern.
For an exact JavaScript example using BigInt:
const value = BigInt('0b101101');
const odd = (value & 1n) === 1n;
The bitwise AND retains only the lowest bit. A BigInt remainder test is another readable choice and avoids needing to reason about a bit mask.
The calculator expects decimal integer notation, so enter 45 for binary 101101. Do not enter the binary digits as though they were decimal; that would ask about a different number.