Explain why 1 +1=0 mod 2

3 upvotes

3 years and 10 months later

GPT-3.5

December 28, 2022

In mathematics, the modulo operation is a way of finding the remainder when one number is divided by another. When we say that 1+1=0 mod 2, we are expressing the fact that the sum of 1 and 1 is congruent to 0 modulo 2. To understand this, it helps to first understand what it means for two numbers to be congruent modulo a third number. Essentially, two numbers are congruent modulo a third number if they have the same remainder when divided by that number. For example, 5 is congruent to 1 modulo 4 because both 5 and 1 leave a remainder of 1 when divided by 4. Now, let's consider the equation 1+1=0 mod 2. This means that when we add 1 and 1 together, the result is congruent to 0 modulo 2. In other words, the sum of 1 and 1 leaves a remainder of 0 when divided by 2. This is indeed the case, as 2 is an even number and therefore leaves a remainder of 0 when divided by 2. Therefore, 1+1=0 mod 2 is a true statement.

GPT-6 Astra

October 19, 2026

Working modulo 2 means keeping only the remainder after dividing by 2.

Since $1+1=2$ , and $2$ has remainder $0$ when divided by $2$ ,

1+1\equiv 0\pmod 2.

Another way to see it: modulo 2, $1$ represents odd numbers and $0$ represents even numbers. Odd + odd = even, so $1+1=0$ in this system.

What changed

Both answers are correct, so the real difference is economy. GPT-3.5 takes four paragraphs to reach a one-line fact. It defines congruence, offers a useful side example (“5 is congruent to 1 modulo 4”), then restates the claim almost word for word before concluding that “1+1=0 mod 2 is a true statement.” It reads like an essay padded to a word count, and the key step, that 1+1 is 2, only surfaces in passing when it notes “2 is an even number.”

Astra covers the same ground in three lines and then adds the idea GPT-3.5 never quite states: modulo 2, “1 represents odd numbers and 0 represents even numbers,” so “Odd + odd = even.” That reframing makes the result feel obvious rather than merely checked. GPT-3.5’s 5 and 1 example does give a beginner something extra to hold onto, but Astra explains more with far fewer words.

Analysis by Claude Opus 5.5, comparing the two answers above.