A person X wants to distribute some pens among six children A, B, C, D, E and F. Suppose A gets twice the number of pens received by B, three times that of C, four times that of D, five times that of E and six times that of F. What is the minimum number of pens X should buy so that the number of pens each one gets is an even number?

The correct option is 294.

Let the number of pens received by children A, B, C, D, E, and F be denoted by $a$, $b$, $c$, $d$, $e$, and $f$ respectively.

According to the given condition, A gets:
- twice the number of B: $a=2b$ ⇒ $b=\frac{a}{2}$
- three times that of C: $a=3c$ ⇒ $c=\frac{a}{3}$
- four times that of D: $a=4d$ ⇒ $d=\frac{a}{4}$
- five times that of E: $a=5e$ ⇒ $e=\frac{a}{5}$
- six times that of F: $a=6f$ ⇒ $f=\frac{a}{6}$

The total number of pens distributed, let's call it T, is:
$T=a+b+c+d+e+f$

Substituting the relations in terms of $a$:
$T=a+\frac{a}{2}+\frac{a}{3}+\frac{a}{4}+\frac{a}{5}+\frac{a}{6}$

To simplify this expression, we find the Least Common Multiple (LCM) of the denominators 2, 3, 4, 5, and 6, which is 60. Taking the common denominator:
$T=\frac{60a+30a+20a+15a+12a+10a}{60}$
$T=\frac{147a}{60}$

Since the number of pens each person gets must be an integer, $a$ must be a multiple of 60 to cancel out the denominator. Let $a=60k$, where $k$ is a positive integer.

If we substitute $a=60k$, the shares of each child become:
- $a=60k$
- $b=30k$
- $c=20k$
- $d=15k$
- $e=12k$
- $f=10k$

We are given the condition that the number of pens each one gets must be an even number. Let us check the shares for different values of $k$:
If $k=1$, then $d=15$, which is an odd number. Therefore, $k$ must be an even integer to ensure that $d=15k$ is even (since 60, 30, 20, 12, and 10 are already even coefficients, their products with any integer are even, but 15 requires an even multiplier).

The smallest positive even integer for $k$ is:
$k=2$

Substituting $k=2$ back into the total pen count equation:
$T=\frac{147}{60}\times 60k=147k$
$T=147\times 2=294$

Thus, the minimum number of pens X should buy is 294.