Answer to Question #106147 in Real Analysis for lessinah

Question #106147
Use a proof of contradiction to establish the following:
If a positive whole number n can be expresses as n1 n2, where n1 is greater equals to 2 and n2 is greater equals to 2, then at least one element sets n1 and n2 is less than n^1/2
1
Expert's answer
2020-03-23T17:10:11-0400

Suppose "n" is positive integer number such that "n=n_1.n_2" where

"n_1 \\geq2 \\ and \\ n_2\\geq2."


Therefore,"n_1\\ and \\ n_2" are two proper divisors other than 1.

If "n_1 \\ or \\ n_2" are equal to 1 then nothing to prove because 1 always fulfills that condition, i.e., "1\\leq n^\\frac{1}{2}" .

Claim":=" At least one of "n_1\\ or \\ n_2" is less than "n^{ \\frac{1}{2}}" .

Now we prove it using the method of contradiction.


On the contrary, assume that "n_1>n^\\frac{1}{2} \\ and \\ n_2>n^\\frac{1}{2}."

Then ,"n=n_1.n_2 >n^\\frac{1}{2}.n^\\frac{1}{2}"

"\\implies" "n>n"

Therefore, we get a contradiction. Hence an assumption was wrong.

Hence the claim was proved by the method of contradiction.



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