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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 x be the smaller of the two consecutive even positive integers. Then, the other integer is x + 2.
Since both the integers are larger than 5,
x > 5 ... (1)
Also, the sum of the two integers is less than 23.
x + (x + 2) < 23
⇒ 2x + 2 < 23
⇒ 2x < 23 – 2
⇒ 2x < 21
⇒ x < `21/2`
⇒ x < 10.5 ...(2)
From (1) and (2), we obtain 5 < x < 10.5.
Since x is an even number, x can take the values 6, 8, and 10.
Thus, the required possible pairs are (6, 8), (8, 10), and (10, 12).
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