Pdirienzo
Is the positive integer n the square of an integer?
(1) 4n is the square of an integer
(2) n^3 is the square of an integer
For n to be a square of an integer, all its distinct prime factors must have an even exponent.
If you are not sure why, check this video
https://youtu.be/DxIH8rjhpKYor these posts
https://anaprep.com/number-properties-f ... -a-number/https://anaprep.com/number-properties-f ... ct-square/Assume \(n = 2^a * 3^b * 5^c\)
For n to be a perfect square, all a, b and c must be even integers.
(1) 4n is the square of an integerSo 4n will have even exponents of all distinct prime factors. 4n is simply \(2^2 * n = 2^2 * 2^a * 3^b * 5^c \)
Here, all (a+2), b and c are given to be even integers. This means that a, b and c are even integers.
Hence, n is a perfect square. Sufficient alone.
(2) n^3 is the square of an integer\(n^3 = (2^a * 3^b * 5^c)^3 = 2^{3a} * 3^{3b} * 5^{3c}\)
We are given that 3a, 3b and 3c are all even integers. Then a, b and c must be even integers too (the 2 must come from a, b and c in 3a, 3b and 3c).
Hence, n is a perfect square. Sufficient alone.
Answer (D)