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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Q. 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

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 $5 \times 5$ , 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 $5 \times 7$ , 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 $12$ .