Discussion

Explanation:

|1 + mn| < |m + n| < 5

Squaring all the expressions

(1 + mn)² < (m + n)² < 25

⇒ 1 + m²n² + 2mn < m² + n² + 2mn
⇒ m² + n² - m²n² – 1 > 0
⇒ (m² – 1) - n²(m² – 1) > 0
⇒ (m² – 1)(1 – n²) > 0
⇒ (m² – 1)(n² – 1) < 0

Case 1: Either m² > 1 and n² < 1
⇒ n = 0 and since |1 + mn| < |m + n| < 5
⇒ 1 < |m| < 5
⇒ m = ±2, ±3 or ±4
∴ 6 possible pairs of (m, n)

Case 2: Either n² > 1 and m² < 1
⇒ m = 0 and since |1 + mn| < |m + n| < 5
⇒ 1 < |n| < 5
⇒ n = ±2, ±3 or ±4
∴ 6 possible pairs of (m, n)

∴ Total 6 + 6 = 12 possible pairs of (m, n)

Hence, 12.

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