To understand the Divisibility Principle of 6 you must know :-
Divisibility Principle of 2 (topic already available in this site)
Divisibility Principle of 3 (topic already available in this site)

Definition :- It involves two step process :-
Step.1)The number has to be divisible by 2.
Step.2)The number has to be divisible by 3.
In simple words, we can say that the given number is divisible by 6, if it is divisible by 2 and 3 both.

Read the following examples and you can further learn Divisibility Principle of 6:-

Example 1 = Is 48 divisible by 6 ?
Answer = Step 1 = Check whether the given number is divisible by 2 or not ?
(Remember; a number is divisible by 2 if has 0, 2, 4, 6, 8 at its one's place.)
Now, the given number has 8 in its ones place,
so we can say that the given number 48 is divisible by 2.
Step 2 = Check whether the given number is divisible by 3 or not ?
(Remember;If the sum of the digits is a multiple of 3,then the number is divisible by 3. )
Now, add the digits of the given number,
4 + 8 = 12
Since 12 is divisible by 3,
so we can say that the given number 48 is divisible by 3 also.
Now, we know that the given number 48 is divisible by both 2 and 3,
Hence, the given number 48 is divisible by 6.

Example 2 = Is 27 divisible by 6 ?
Answer = Step 1 = Check whether the given number is divisible by 2 or not ?
(Remember; a number is divisible by 2 if has 0, 2, 4, 6, 8 at its unit/ones place.)
Now, the given number has 7 in its ones place,
so we can say that the given number 27 is not divisible by 2, and step 2 not required,
Hence, the given number 27 is not divisible by 6.

Example 3 = Is 16 divisible by 6 ?
Answer = Step 1 = Check whether the given number is divisible by 2 or not ?
(Remember; a number is divisible by 2 if has 0, 2, 4, 6, 8 at its unit/ones place.)
Now, the given number has 6 in its ones place,
so we can say that the given number 48 is divisible by 2.
Step 2 = Check whether the given number is divisible by 3 or not ?
(Remember;If the sum of the digits is a multiple of 3, then the number is divisible by 3. )
Now, add the digits of the given number,
1 + 6 = 7
Since 7 is not divisible by 3,
so we can say that the given number 16 is not divisible by 3.
Now, we know that the given number 16 is not divisible by both 2 and 3,
Hence, the given number 16 is not divisible by 6.

Example 4 = Is 216 divisible by 6 ?
Answer = Step 1 = Check whether the given number is divisible by 2 or not ?
(Remember; a number is divisible by 2 if has 0, 2, 4, 6, 8 at its unit/ones place.)
Now, the given number has 6 in its ones place,
so we can say that the given number 216 is divisible by 2.
Step 2 = Check whether the given number is divisible by 3 or not ?
(Remember;If the sum of the digits is a multiple of 3, then the number is divisible by 3. )
Now, add the digits of the given number,
2 + 1 + 6 = 9
Since 9 is divisible by 3,
so we can say that the given number 216 is divisible by 3 also.
Now, we know that the given number 216 is divisible by both 2 and 3,
Hence, the given number 216 is divisible by 6.

Example 5 = Is 543 divisible by 6 ?
Answer = Step 1 = Check whether the given number is divisible by 2 or not ?
(Remember; a number is divisible by 2 if has 0, 2, 4, 6, 8 at its unit/ones place.)
Now, the given number has 3 in its ones place,
so we can say that the given number 543 is not divisible by 2, and so step 2 not required,
Hence, the given number 543 is not divisible by 6.

Example 6 = Is 806 divisible by 6 ?
Answer = Step 1 = Check whether the given number is divisible by 2 or not ?
(Remember; a number is divisible by 2 if has 0, 2, 4, 6, 8 at its unit/ones place.)
Now, the given number has 6 in its ones place,
so we can say that the given number 806 is divisible by 2.
Step 2 = Check whether the given number is divisible by 3 or not ?
(Remember;If the sum of the digits is a multiple of 3, then the number is divisible by 3. )
Now, add the digits of the given number,
8 + 0 + 6 = 14
Since 14 is not divisible by 3,
so we can say that the given number 806 is not divisible by 3.
Now, we know that the given number 806 is not divisible by both 2 and 3,
Hence, the given number 806 is not divisible by 6.