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What is the least possible positive integer-value of N such that \(\sqrt{18 \cdot n \cdot 34}\) is an integer?

 Jul 13, 2019
 #1
avatar+8842 
+5

\(\phantom{=\quad}\sqrt{18\cdot n\cdot34}\\~\\ {=\quad}\sqrt{3\cdot3\cdot2\cdot n\cdot2\cdot17}\\~\\ {=\quad}\sqrt{3\cdot3\cdot2\cdot 2\cdot17\cdot n}\\~\\ {=\quad}\sqrt{3\cdot3}\cdot\sqrt{2\cdot 2}\cdot\sqrt{17\cdot n}\\~\\ {=\quad}3\cdot2\cdot\sqrt{17\cdot n}\\~\\ {=\quad}6\sqrt{17n}\)

 

There must be a pair of  17's  in the prime factorization of the number under the sqrt.

 

So  17  must be a factor of  n  , and the least possible positive integer value of  n  is  17

 

\(\phantom{=\quad}6\sqrt{17\cdot17}\\~\\ {=\quad}6\cdot17\\~\\ {=\quad}102\)_

 Jul 13, 2019
 #2
avatar+2551 
+2

In these problems, simplify the radical so its easier to understand.

CalculatorUser  Jul 13, 2019

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