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Sum of Two Integers LC #371 Medium XOR · Carry
Problem

Given two integers a and b, return the sum of the two integers without using the operators + and -.

Example 1
Input: a = 1, b = 1
Output: 2
Explanation: 1 + 1 = 2 using XOR + carry.
Example 2
Input: a = 2, b = 3
Output: 5
Explanation: XOR gives partial sum; AND shifted left gives carry.
Constraints: -1000 ≤ a, b ≤ 1000
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Approach
XOR + Carry Loop
XOR gives bit-sum without carry. AND finds carry positions. Shift carry left and repeat until carry is zero (b=0). Each iteration propagates carries one position left.
Binary Representation
Variables
a
—
b
—
carry
—
iter
0
Step Logic
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Algorithm
1
XOR (a^b) computes bit-sum without carry
2
AND (a&b) finds positions where carry occurs
3
Shift carry left by 1: carry << 1
4
Repeat until b=0 (no more carry to add)
Time
O(log n)
Space
O(1)
Why It Works

Binary addition works column by column. XOR at each bit gives the sum-without-carry. AND at each bit gives where two 1s overlap — that's the carry. Shifting the carry left by 1 places it in the correct next column. We repeat until there is no more carry (b becomes 0). This exactly mimics how hardware adders work.