What is the least possible positive integer-value of N such that \(\sqrt{18 \cdot n \cdot 34}\) is an integer?
+9466 \(\phantom{=\quad}\sqrt{18\cdot n\cdot34}\\~\\ {=\quad}\sqrt{3\cdot3\cdot2\cdot n\cdot2\cdot17}\\~\\ {=\quad}\sqrt{3\cdot3\cdot2\cdot 2\cdot17\cdot n}\\~\\ {=\quad}\sqrt{3\cdot3}\cdot\sqrt{2\cdot 2}\cdot\sqrt{17\cdot n}\\~\\ {=\quad}3\cdot2\cdot\sqrt{17\cdot n}\\~\\ {=\quad}6\sqrt{17n}\)
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
\(\phantom{=\quad}6\sqrt{17\cdot17}\\~\\ {=\quad}6\cdot17\\~\\ {=\quad}102\)_
+2862 In these problems, simplify the radical so its easier to understand.
