What is the smallest positive integer n so that 3n squared is a multiple of 4

n=2

(3*2)^2=6^2=36

36=9*4

if n is even then n^2 is divisible by 4

so we need the smallest even integer which is 2

check 3n^2 = 3(2^2) = 12 , which is a multiple of 4

(I read your expression as 3 n^2
if you mean (3n)^2, then the answer is still the same )

the smallest integer n is 2

To find the smallest positive integer, n, such that 3n^2 is a multiple of 4, we can follow these steps:

Step 1: Understand the problem
We need to find the smallest value of n for which 3n^2 is divisible evenly by 4.

Step 2: Identify the requirements
A number is divisible by 4 if it is divisible by 2 twice. Hence, we need to find the smallest n for which 3n^2 is divisible by 2 twice.

Step 3: Represent the problem mathematically
We can express the divisibility of 3n^2 by 4 using the modulo operator (%):
(3n^2) % 4 = 0

Step 4: Simplify the expression
We can simplify the equation (3n^2) % 4 = 0 further:
(3n^2 % 4) = (0 % 4)
(3n^2 % 4) = 0

Step 5: Find the values of n
We need to find the value of n that satisfies the equation (3n^2) % 4 = 0.

Step 6: Test values of n
We can start testing values of n until we find the smallest n that satisfies the equation.

For n = 1:
(3 * 1^2) % 4 = 3 % 4 = 3
Since 3 is not divisible evenly by 4, n = 1 does not satisfy the condition.

For n = 2:
(3 * 2^2) % 4 = 12 % 4 = 0
Here, 12 is divisible evenly by 4, so n = 2 satisfies the condition.

Therefore, the smallest positive integer n that makes 3n^2 a multiple of 4 is n = 2.