Possible Values of x for 3x 12

Introduction: In this problem, we need to find the possible values of x if 3x+12 is perfectly divisible by a positive integer x.

Step 1: Express 3x+12 as a product of x and another integer k.

3x+12 = x(k)

Factor out 3 from the left side:

3(x+4) = x(k)

Step 2: Since k is an integer, x+4 must be divisible by 3.

If x+4 is divisible by 3, then x can be expressed as:

x = 3n-4, where n is a positive integer.

Substituting this value of x in the expression for 3x+12:

3x+12 = 3(3n-4)+12 = 9n

So, x is a factor of 9n.

Step 3: Let's consider the factors of 9n.

The factors of 9n are 1, 3, 9, n, 3n, and 9n.

Step 4: We need to eliminate the factors that are not possible values of x.

- If x = 1, then 3x+12 = 15, which is not divisible by x.
- If x = 3, then 3x+12 = 21, which is not divisible by x.
- If x = 9, then 3x+12 = 39, which is not divisible by x.

Step 5: The remaining factors are possible values of x.

- If x = n, then 3x+12 = 3n+12 = 3(n+4), which is divisible by x.
- If x = 3n, then 3x+12 = 9n+12 = 3(3n+4), which is divisible by x.
- If x = 9n, then 3x+12 = 27n+12 = 3(9n+4), which is divisible by x.

Therefore, the possible values of x are n, 3n, and 9n, where n is a positive integer.

Step 6: Conclusion

Since there are 3 possible values of x for each value of n, the total number of possible values of x is 3 times the number of positive integers n. As there are infinite positive integers, there are infinite possible values of x.