Significance of fundamental theorem of arithmetic — lesson. Mathematics State Board, Class 10.

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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 \(5 \times 6\).

\frac{5 \times 6}{3}

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 \(2 \times 6\).

\frac{2 \times 6}{4}

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:

\(xyxyxy = (xy \times 10000) + (xy \times 100) + xy\)

\(xyxyxy = xy (10000 + 100 + 1)\)

\(xyxyxy = xy (10101)\)

\frac{xyxyxy}{10101} = 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 \(10101\).

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2. Significance of fundamental theorem of arithmetic