Theory:

1. If a prime number \(p\) divides \(ab\), then \(p\) divides either \(a\) or \(b\). That is, \(p\) divides at least one of them.
Example:
Let us take a prime number \(3\) divides \(5 \times 6\).
 
5×63
 
Here, \(3\) cannot divide \(5\), but it divides \(6\).
 
That is, a prime number \(p\) divides at least one of them.
 
2. If a composite number \(n\) divides \(ab\), then \(n\) neither divide \(a\) nor \(b\).
Example:
Let us take a  composite number \(4\) divides \(2 \times 6\).
 
2×64
 
Here, \(4\) neither divide \(2\) nor divide \(6\). But, it divides the product of \(2 \times 6 = 12\).
 
Thus, if a composite number \(n\) divides \(ab\), then \(n\) neither divide \(a\) nor \(b\).
Fun Fact
The six-digit number of the form \(xyxyxy\) (where \(1 \le x \le 9, 1 \le y \le 9\)) always divisible by the number \(10101\).
 
Explanation:
 
\(xyxyxy = (xy \times 10000) + (xy \times 100) + xy\)
 
\(xyxyxy = xy (10000 + 100 + 1)\)
 
\(xyxyxy = xy (10101)\)
 
xyxyxy10101=xy
 
Thus, any six-digit number of the form \(xyxyxy\) (where \(1 \le x \le 0, 1 \le y \le 9\)) always divisible by the number \(10101\).