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Given that the polynomial \(x^2 - kx + 60\) has only positive integer roots, find the average of all distinct possibilities for k.

 May 3, 2022
 #1
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I hope this is correct.

k can be 61, 32, 23, 19, 17, or 16. When we add and divide by 6, we get 28.

 May 3, 2022
 #2
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By Vieta's formula, the product of roots of \(x^2 - kx + 60\) is 60.

 

What integers multiply to 60?

 

160
230
320
415
512
610
106
125
154
203
302
601

 

Each row represents the two roots of a specific value of k.

The sum of roots is k by Vieta's formula again. Therefore, you can find the possibilities for the sum of roots, then find the average.

 May 3, 2022
edited by MaxWong  May 3, 2022

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