Answer to Question #106146 in Real Analysis for lessinah

Question #106146
Use a proof bt of the set of n1 and n2 is less than n^1/2y 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
2020-03-24T13:22:06-0400

"n=n_1n_2," "n_1\\geq2 , n_2\\geq2" .

Use a proof by contradiction. Let us assume that both of "n_1,\\,n_2" are greater than or equal to "n^{1\/2}."

Case I

"n_1=n^{1\/2}"

"n_2=n^{1\/2}"

"n_1n_2=n^{\\frac12+\\frac12}=n"

Case II

"n_1=n^{\\frac12+x}; x>0"

"n_2=n^{\\frac12+y}; y>0"

"n_1n_2=n^{\\frac12+x}n^{\\frac12+y}; x>0 ; y>0"

"n_1n_2=nn^{x+y}; x>0 ; y>0"

So, "x+y=0"

"x=-y" but "x>0,y>0"

which is a contradiction.

So,If a positive whole number "n" can be expressed as "n_1 n_2" , where "n_1" is greater than or equal to 2 and "n_2" is greater than or equal to 2, then at least one element of "n_1" and "n_2" is less than "n^{1\/2}" .



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