Number Theory least divisor of integer is prime number if integer is not prime

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SUMMARY

The discussion centers on the statement that the least divisor (excluding 1) of a non-prime integer is always a prime number. Participants provided examples such as 8 (least divisor 2), 10 (least divisor 2), 9 (least divisor 3), and 121 (least divisor 11), all confirming that the least divisor is indeed prime. The challenge lies in generalizing this observation for all non-prime integers, as non-prime divisors can be further divided. The consensus is that the statement holds true based on the provided examples.

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Homework Statement


The question is not really a question from a book but rather a statement that it makes : it says " Obviously the least divisor[excluding 1] of an integer a is prime if a itself is not prime." I kind of believe this statement but I'm having trouble proving the general case

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The Attempt at a Solution


when I take a few examples : a =8 , 2 (LD) is prime . a = 10, 2(LD) . a = 9, 3(LD) is prime a =121, 11(LD) is prime. But I'm having trouble generalizing this for all n.
 
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any non-prime divisor can be further divided.
 
Thank you! I feel stupid but happy I understand now.
 

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