+12 What is the smallest positive integer n such that root(4)(1323*n) is an integer?
+1765 The fourth root of a number is only an integer if the prime factorization of the number contains every prime factor raised to a power that is a multiple of 4.
In this case, we need to analyze the prime factorization of 1323=3×7×7.
3 is already raised to the power of 1 (which is a multiple of 4).
7 needs to be raised to a power that is a multiple of 4. Currently, it's raised to the power of 2.
Therefore, the smallest value of n that makes 1323n have a fourth root as an integer is the one that raises 7 to the fourth power. This means n=72=49.
With n=49, 1323n=1323×49=64809, and its fourth root is 464809=3×7=21, which is an integer.
Therefore, the answer is 49.
