# How many factors does 144 have?

Mar 30, 2016

$15$ including $1$.

#### Explanation:

Factors of $144$ are {1,2,3,4,6,8,9,12,16,18,24,36,48,72,144}

i.e. in all $15$ factors including $1$.

Mar 30, 2016

$15$

#### Explanation:

To find the answer, first factor $144$ into prime factors:

$144 = 2 \times 72$

$= 2 \times 2 \times 36$

$= 2 \times 2 \times 2 \times 18$

$= 2 \times 2 \times 2 \times 2 \times 9$

$= 2 \times 2 \times 2 \times 2 \times 3 \times 3$

$= {2}^{4} \times {3}^{2}$

Any positive factor of $144$ can be expressed as:

${2}^{a} \times {3}^{b}$

where $a = 0 , 1 , 2 , 3 , 4$ and $b = 0 , 1 , 2$

That gives $5$ possible values for $a$ and $3$ possible values for $b$ and therefore $5 \times 3 = 15$ factors.

Since there are only two distinct primes involved, we can write the factors in a grid with columns for the distinct powers of $2$ and rows for the distinct powers of $3$:

$\textcolor{w h i t e}{000} 1 \textcolor{w h i t e}{000} 2 \textcolor{w h i t e}{000} 4 \textcolor{w h i t e}{000} 8 \textcolor{w h i t e}{00} 16$

$\textcolor{w h i t e}{000} 3 \textcolor{w h i t e}{000} 6 \textcolor{w h i t e}{00} 12 \textcolor{w h i t e}{00} 24 \textcolor{w h i t e}{00} 48$

$\textcolor{w h i t e}{000} 9 \textcolor{w h i t e}{00} 18 \textcolor{w h i t e}{00} 36 \textcolor{w h i t e}{00} 72 \textcolor{w h i t e}{0} 144$

May 2, 2017

it has 15 factors ={ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144}

#### Explanation:

$144 = 1 \cdot 144$
$144 = 2 \cdot 72$
$144 = 3 \cdot 48$
$144 = 4 \cdot 36$
$144 = 6 \cdot 24$
$144 = 8 \cdot 18$
$144 = 9 \cdot 16$
$144 = 12 \cdot 12$

it has 15 factors ={ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144}

May 2, 2017

15

#### Explanation:

there is another for you to observe @@
$144 = {2}^{4} \cdot {3}^{2}$
5 possibilities to choose 2^(0~4)
3 possibilities to choose 3^(0~2)#
so
$\left(4 + 1\right) \cdot \left(2 + 1\right) = 15$

May 2, 2017

A = $15$

#### Explanation:

The most accurate, yet the most complex method is first, prime decomposing and then using the exponent to calculate how many factors a set number $x$ has! Furthermore, this method is extremely useful when calculating larger numbers rather than having to list out every single factor - a tiresome and boring process.

1. Prime decompose the number:

= $144$
= ${2}^{4} \cdot {3}^{2}$

1. Add $1$ to each exponent:

= ${2}^{4 + 1} \cdot {3}^{2 + 1}$
= ${2}^{5} \cdot {3}^{3}$

1. Find the products of the exponents:

= $5 \cdot 3$
= $15$

Best of luck!