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.