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