An angle measuring (468n)° is in standard position. for which value of n will the terminal side fall along the negative portion of the x-axis?
n = 4
n = 5
n = 6
n = 7

The correct answer is:

n = 5.

Explanation:

On a coordinate grid, an angle starts on the positive x-axis, at 0°.  Multiples of 360° will have their terminal side here as well.

If the terminal side of an angle is on the negative x-axis, this means it is a multiple of 180°.

This gives us 468n° and 180°.  We want the LCM of these two.  To find this, we first find the prime factorization of each number:

468 = 2*234
234 = 2*117
117 = 3*39
39 = 3*13

This gives us:
468 = 2*2*3*3*13
468 = 2²(3²)(13)

We now find the prime factorization of 180:
180 = 2*90
90 = 2*45
45 = 5*9
9 = 3*3

This gives us:
180 = 2*2*3*3*5
180 = 2²(3²)(5)

For the LCM, we multiply the common factors by the uncommon.  The common factors of 468 and 180 are:
2²(3²)

The uncommon factors are:
5*13

This gives us an LCM of:
2²(3²)(5)(13) = 2340

To find the value of n, we solve the equation:
468n = 2340

Divide both sides by 468:
468n/468 = 2340/468
n = 5

The answer is n=5. 

I got the answer through solving all ns and graphing them.

If n = 4 -> 468 (4) = 1848 degrees. It falls on the first quadrant, thus it does not fall along the negative portion of the x-axis. 

If n=5 -> 468 (5) = 2310 degrees. It falls on the second quadrant, thus it falls along the negative portion of the x-axis.

If n=6 -> 468 (6) = 2808 degrees. It falls on the fourth quadrant, thus it does not fall along the negative portion of the x-axis.

If n=7 -> 468 (7) = 3276 degrees. It falls on the first quadrant, thus it does not fall along the negative portion of the x-axis.