It's a good problem on the concept that the complete angle in a circle is 360 degrees.
First plate has 4 holes and hence correspondingly has 4 hole-centres. All of these hole-centres are uniformly spaced at an angle of 360/4 = 90 degrees
Similarly,5 hole-centres of the second plate are uniformly spaced at an angle of 360/5 = 72 degrees.
First a hole of one plate is aligned with a hole of another plate.
After this arrangement,just visualise the positions of the holes of the two plates.
The angular spacing between the two nearest holes of two plates = 90-72 = 18 degrees
So if we move one of the plates by 18 degrees clockwise or anti-clockwise depending on which plate we are moving, then a new pair of holes will be aligned.
Answer "C"

sam117 wrote:

OA: C

Please see the image. Does anyone have an idea how to solve this?

Let's call the 4-holed plate ABCD, with holes A, B, C and D.
Since 360/4 = 90, the holes are spaced 90° apart.
Let A=0°, B=90°, C=180°, and D=270°.

Let's call the 5-holed plate RSTUV, with holes R, S, T, U, and V.
Since 360/5 = 72, the holes are spaced 72° apart.
Let R=0°, S=72°, T=144°, U=216°, and V=288°.

Let the plates be positioned so that A is aligned with R at 0°.

Since S is at 72° and B is at 90° -- a difference of 18° -- S will align with B if plate RSTUV rotates 18° degrees clockwise or if plate ABCD rotates 18° counter-clockwise.
There is no difference smaller than 18°.
Thus, the least number of degrees that one of the plates must rotate is 18°.

The correct answer is C.