Three positive integers x, y and z are in arithmetic progression. If y − x > 2 and xyz = 5(x + y + z), then z − x equals
Explanation:
Let the three integers x, y and z be (a - d), a, (a + d) respectively.
[(a – d), a and (a + d) are all positive integers]
Since y – x > 2, hence d > 2.
Given, xyz = 5(x + y + z)
⇒ (a – d) × a × (a + d) = 5 × 3a
⇒ (a – d)(a + d) = 15
Here (a – d) and (a + d) are positive integers
∴ We need to write 15 as product of 2 positive integers. This can be done in two ways, 1 × 15 or 3 × 5
Hence, (a, d) is either (8, 7) or (4, 1).
Since d > 0 hence, (4, 1) is rejected.
∴ (a, d) = (8, 7)
∴ z – x = (a + d) – (a - d) = 2d = 14
Hence, option (c).
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