PDF chapter test TRY NOW
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 \(5 \times 6\).
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 divides \(a\) nor \(b\).
Example:
Let us take a composite number \(4\) divides \(2 \times 6\).
Here, \(4\) neither divides \(2\) nor divides \(6\). But, it divides the product of \(2 \times 6 = 12\).
Thus, if a composite number \(n\) divides \(ab\), then \(n\) neither divides \(a\) 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:
\(xyxyxy = (xy \times 10000) + (xy \times 100) + xy\)
\(xyxyxy = xy (10000 + 100 + 1)\)
\(xyxyxy = xy (10101)\)
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 \(10101\).