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April 11th, 2018, 05:29 AM   #1
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How can you find the factors of a number easily?

How can you find the factors of a number easily? It is probably by prime factorisation, but I am not sure how prime factorisation can be used to find factors. Please help!

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April 11th, 2018, 06:28 AM   #2
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How can you find the factors of a number easily? It is probably by prime factorisation, but I am not sure how prime factorisation can be used to find factors. Please help!

Thanks!
First, try dividing by 2 as many times as it will divide.

Then try dividing by 3, by 5, by 7, etc.

For example:756.
$\displaystyle 756 \div 2 = 378$

$\displaystyle 378 \div 2 = 189$

$\displaystyle 189 \div 2 = 94.5$

*So we have 2 factors of 2.

$\displaystyle 189 \div 3 = 63$

$\displaystyle 63 \div 3 = 21$

$\displaystyle 21 \div 3 = 7$

$\displaystyle 7 \div 3 = 2.333333$

*So we have 3 factors of 3.

$\displaystyle 7 \div 5 = 1.4$

*So we have no factors of 5.

$\displaystyle 7 \div 7 = 1$

*So we have 1 factor of 7.

Combing the statements in * we have that $\displaystyle 756 = 2^2 \cdot 3^3 \cdot 7$

-Dan
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April 11th, 2018, 06:32 AM   #3
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How can you find the factors of a number easily? It is probably by prime factorisation, but I am not sure how prime factorisation can be used to find factors. Please help!

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You are correct.

To find all possible factors of an integer, start by doing a prime factorization of the absolute value of the integer in question. Then take 1, those prime factors, and all possible multiples of those factors both plus and minus to find all possible factors.

Example. 196 = 2 * 98 = 2 * 2 * 49 = 2 * 2 * 7 * 7.

So the possible factors are

$\pm 1,\ \pm 2,\ \pm 7,\ \pm 4,\ \pm 14,\ \pm 49,\ \pm 28,\ \pm 98,\ \pm 196.$
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