Divisibility criteria of 24. Why is this? I am currently familiar with the method of checking if a number is divisible by $2, 3, 4, 5, 6, 8, 9, 10, 11$. While Checking for divisibility for $24$ (online). I found out that the number has to satisfy the divisibility criteria of $3$ and $8$. I agree this gives the answer. But why cant I check the divisibility using the divisibility criteria of $6$ and $4$ ? Is there a rule to this criteria ?
 A: If $3 \mid n$ and $8 \mid n$, then clearly the $\mathrm{lcm}(3,8) = 24 \mid n$, since the least common multiple of $a$ and $b$ clearly divides any number divisible by both $a$ and $b$.
On the other hand, $4 \mid n$ and $6 \mid n$ is only enough to conclude that $\mathrm{lcm}(4,6) = 12 \mid n$.
A: If you are divisible by both $6$ and $4$, the number could be 12 which is not divisible by 24.
The reason for using $3$ and $8$ is that the least common multiple is 24. So every number that is divisible by both $3$ and $8$ is divisible by $24$. 
A: It is not true that if a number's divided by $\,6\,,\,4\,$ then it is divided by $\,6\cdot 4\,$ , as $\,12\,$ proves. Yet it is true that if a number's divided by $\,3\,,\,8\,\,$ then it i divided by $\,3\cdot 8=24\,$. Why? Because the former pair is not coprime (i.e., its minimal common divisor is not $\,1\,$), whereas the latter pair is coprime...Can you take it from here?
A: For instance, $12$ is divisible by $6$ and $4$ but is not divisible by $24$. The problem with $6$ and $4$ is that the $\gcd(6,4) = 2 \neq 1$. You could split $24$ as $8$ and $3$ and check for divisibility by $8$ and $3$ because $\gcd(3,8) = 1$.
In general, to check for divisibility by $n$, look at the prime decomposition of $n = p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ and check whether the number is divisible by $p_j^{\alpha_j}$ for all $j \in \{1,2,\ldots,k\}$.
Another equvialent way is to write $n = ab$ such that $\gcd(a,b) = 1$. Then a number is divisible by $n$ if and only if the number is divisible by $a$ and $b$.
A: The problem here might be something like $12$. You see, we have that $12$ is divisible by both $6$ and $4$, but it's not divisible by $24$. The reason they suggest $3$ and $8$ is because they are relatively prime, meaning that you can't have the sort of overlap in the case of $6$ and $4$.
This all has to do with the Fundamental Theorem of Arithmetic, which says that each number can be written uniquely as a product of primes, and primes have the special characteristic (or as Marvis points out, they are defined to be exactly those numbers with the characteristic) that if $p|ab$, then $p|a$ or $p|b$. So if $3$ and $8$ divide a number, then $24$ divides that number. But $6$ and $4$ dividing a number doesn't even guarantee that $8$ divides that number.
A: According to cross divisibility test (VJ's universal divisibility test) there are infinite test for any number. For 24  the divisibility test are given as 
 1) 24 | (10T+U) if and only if 24 | (2T+ 5U )
 2) 24 | (10T+U) if and only if 24 | (2T -7U)
 3) 24 | (10T+U) if and only if 24 | (2T-17U ) etc
To discover why it works refer 'Modern Approach to speed math Secret' at tinyurl.com/mlxk8pw
A: If LCM ( a,b) = n, then any number which is divisible by n is also divisible by a & b because n is least common multiple of a & b.
Now if we want to check divisibility of 24 then we have to follow these steps.


*

*We know that there is no predetermined divisibility rule for 24 as in the case of 2,3,4,5,6,7,8,9,11. So we need to find factors of 24 between 2 to 11.

*Now we have factors of 24 between ( 2 to 11) is   2,3,4,6,8 

*Now find factors whose LCM is 24 ( see above rule).
LCM ( 8, 3) = 24 , from above rule any number divisible by 24 also divisible by 8 & 3 so check the divisibility of that number from 8 & 3 .
We can't consider (4,6) , ( 8,2) or (2,3,4) etc as their LCM is not 24.
Further, there is one more trick to solve this, see below
The LCM of co-prime numbers is product of these numbers because co-prime number has no common factor. Suppose p & q are co-prime numbers then LCM (p,q) = p*q 
Now we can easily find co-prime numbers from factors of 24 ( 2,3,4,6,8) whose multiplication is 24  i.e 8 & 3
8 & 3 are co-prime and its product is 24. 
