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Question
Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.
Solution
Let 2n, 2n + 2 be two positive consecutive integers where n ≥ 1 ∈ Z.
Given that
2n > 5 and 2n + 2 > 5
∴ `"n">5/2` and 2n > 2
∴ `"n">5/2` and `"n">3/2`
∴ `"n">5/2` ......(i)
Also (2n) + (2n + 2) < 23
∴ 4n + 2 < 23
∴ 4n < 21
∴ `"n"<21/4` .....(ii)
From (i) and (ii)
`5/2<"n"<21/4` and n is an integer.
∴ n = 3, 4, 5
n = 3 gives 2n = 6, 2n + 2 = 8
n = 4 gives 2n = 8, 2n + 2 = 10
n = 5 gives 2n = 10, 2n + 2 = 12
∴ The pairs of positive even consecutive integers are (6, 8) (8, 10), (10, 12)
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