If X and Y are odd, which expression must be odd?

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Multiple Choice

If X and Y are odd, which expression must be odd?

Explanation:
When a number is odd, it can be written as 2n+1. If X and Y are odd, let X = 2a+1 and Y = 2b+1. Then: - X − Y = (2a+1) − (2b+1) = 2(a−b), which is even. - X + Y = (2a+1) + (2b+1) = 2(a+b+1), which is even. - X − Y + 2 = 2(a−b) + 2 = 2(a−b+1), which is even. Adding 1 to the even result flips the parity: X − Y + 1 = 2(a−b) + 1, which is odd. So the expression that must be odd is X − Y + 1.

When a number is odd, it can be written as 2n+1. If X and Y are odd, let X = 2a+1 and Y = 2b+1. Then:

  • X − Y = (2a+1) − (2b+1) = 2(a−b), which is even.

  • X + Y = (2a+1) + (2b+1) = 2(a+b+1), which is even.

  • X − Y + 2 = 2(a−b) + 2 = 2(a−b+1), which is even.

Adding 1 to the even result flips the parity: X − Y + 1 = 2(a−b) + 1, which is odd.

So the expression that must be odd is X − Y + 1.

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