Directions: Find the number that should replace the question mark in the series.

?, 105, 420, 1260, 6300, 25200, 75600

The correct answer is 21.

To find the missing number that should replace the question mark in the series, let us analyze the mathematical relationship between consecutive terms starting from the known numbers:

$105\times 4=420$

$420\times 3=1260$

$1260\times 5=6300$

$6300\times 4=25200$

$25200\times 3=75600$

By inspecting the sequence of multipliers (4, 3, 5, 4, 3), we can clearly see a repeating cyclic pattern of multiplication factors: 5, 4, 3, 5, 4, 3.

Working backward to find the first term (represented by ?), the operation used to obtain the second term (105) must be multiplication by 5:

$?\times 5=105$

Solving for the unknown value:

$?=\frac{105}{5}=21$

Therefore, the number that replaces the question mark is 21.