If n is a positive integer such that n^2 - 7n + 17 is equal to the product of two consecutive odd integers, find the sum of these odd integers

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Denote Odd Integers: Let's denote the two consecutive odd integers as x and x + 2. Their product is x(x + 2). We need to set up an equation where the product of these two integers is equal to n^2 - 7n + 17. So, we have x(x + 2) = n^2 - 7n + 17.Set Up Equation: Now we need to expand the left side of the equation to get a quadratic equation in terms of x. x(x + 2) = x^2 + 2x.Expand Left Side: We rewrite the equation with the expanded form: x^2 + 2x = n^2 - 7n + 17.Factor Right Side: Since we are looking for integer solutions for x, we can try to factor the right side of the equation to match the left side. However, we notice that the equation is already in terms of n, and we are given that n is a positive integer. Therefore, we need to find values of n for which n^2 - 7n + 17 is a product of two consecutive odd integers.Check Small Values: We can start by checking for small values of n and see if n^2 - 7n + 17 yields a product of two consecutive odd integers. We can do this by calculating the value of n^2 - 7n + 17 for different values of n and checking if the result is a product of two consecutive odd integers.Check n=1: Let's start with n = 1: (1)^2 - 7(1) + 17 = 1 - 7 + 17 = 11, which is not a product of two consecutive odd integers.Check n=2: Now let's try n = 2: (2)^2 - 7(2) + 17 = 4 - 14 + 17 = 7, which is also not a product of two consecutive odd integers.Check n=3: Let's try n = 3: (3)^2 - 7(3) + 17 = 9 - 21 + 17 = 5, which is not a product of two consecutive odd integers.Check n=4: Now let's try n = 4: (4)^2 - 7(4) + 17 = 16 - 28 + 17 = 5, which is not a product of two consecutive odd integers.Check n=5: Let's try n = 5: (5)^2 - 7(5) + 17 = 25 - 35 + 17 = 7, which is not a product of two consecutive odd integers.Check n=6: Now let's try n = 6: (6)^2 - 7(6) + 17 = 36 - 42 + 17 = 11, which is not a product of two consecutive odd integers.Check n=7: Let's try n = 7: (7)^2 - 7(7) + 17 = 49 - 49 + 17 = 17, which is not a product of two consecutive odd integers.Check n=8: Now let's try n = 8: (8)^2 - 7(8) + 17 = 64 - 56 + 17 = 25, which is 5 * 5, but 5 and 5 are not consecutive odd integers.Check n=9: Let's try n = 9: (9)^2 - 7(9) + 17 = 81 - 63 + 17 = 35, which is 5 * 7, and 5 and 7 are consecutive odd integers.Find Solution: We have found that for n = 9, n^2 - 7n + 17 equals 35, which is the product of the consecutive odd integers 5 and 7. Therefore, the sum of these odd integers is 5 + 7.Calculate Sum: The sum of the two consecutive odd integers 5 and 7 is 12.

If $n$ is a positive integer such that $n^2 - 7n + 17$ is equal to the product of two consecutive odd integers, find the sum of these odd integers

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If nnn is a positive integer such that n2−7n+17n^2 - 7n + 17n2−7n+17 is equal to the product of two consecutive odd integers, find the sum of these odd integers

If $n$ is a positive integer such that $n^2 - 7n + 17$ is equal to the product of two consecutive odd integers, find the sum of these odd integers