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Problem #48 · September 13, 2026

Absolute value

For the equation ∣x∣+∣y∣=3|x| + |y| = 3, how many pairs of integers (x; y)(x;\ y) satisfy it?
  1. A3
  2. B4
  3. C8
  4. D12
  5. E16

WORKED EXAMPLE

Solution

∣x∣|x| can only take the values 0,1,2,30, 1, 2, 3. When ∣x∣=0|x| = 0, we have y=±3y = \pm3 (2 pairs); when ∣x∣=3|x| = 3, we have x=±3,y=0x = \pm3, y = 0 (2 pairs); when ∣x∣=1|x| = 1, we have x=±1,y=±2x = \pm1, y = \pm2 (4 pairs); when ∣x∣=2|x| = 2, we have x=±2,y=±1x = \pm2, y = \pm1 (4 pairs). Total: 2+2+4+4=122 + 2 + 4 + 4 = 12 pairs.
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