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Proof Practice-I

\(\text {Proof the following}\)

\(\bullet\) Given any positive integer \(\displaystyle k\) ,prove that there are \(\displaystyle k\) consecutive integers that are all composite.

Note by Anik Mandal
2 years, 10 months ago

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One line proof. \( (k+1)! + i \) where \( 2 \leq i \leq k+1 \)

Siddhartha Srivastava - 2 years, 10 months ago

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Sorry,I didn't get it.Please explain it clearly@Siddhartha Srivastava

Anik Mandal - 2 years, 10 months ago

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Expanding on what Dinesh Chavan said. \( (k+1)! = (k+1)*k*(k-1)...*4*3*2 \). So for each \( 2 \leq i \leq k+1 \), we see that \( i|(k+1)! \). Also, it is trivially true that \( i|i \). Therefore, \( i| (k+1)! + i \). Since \( i \) divides the number and it is greater than one, the number must be composite.

Siddhartha Srivastava - 2 years, 10 months ago

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If u know factorial notation then notice that for each $i$ that same $i$ from factorial expansion will come out common and the expression will be composite.

Dinesh Chavan - 2 years, 10 months ago

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