Question

How many factors does 144 have?

Answer 1

`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`.

Answer 2

`15`

Explanation:

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

`144=2 xx 72`

`=2 xx 2 xx 36`

`=2 xx 2 xx 2 xx 18`

`=2 xx 2 xx 2 xx 2 xx 9`

`=2 xx 2 xx 2 xx 2 xx 3 xx 3`

`=2^4 xx 3^2`

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

`2^a xx 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 xx 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`:

`color(white)(000)1color(white)(000)2color(white)(000)4color(white)(000)8color(white)(00)16`

`color(white)(000)3color(white)(000)6color(white)(00)12color(white)(00)24color(white)(00)48`

`color(white)(000)9color(white)(00)18color(white)(00)36color(white)(00)72color(white)(0)144`

Answer 3

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

Explanation:

`144 = 1 * 144`
`144 = 2 * 72`
`144 = 3 * 48`
`144 = 4 * 36`
`144 = 6 * 24`
`144 = 8 * 18`
`144 = 9 * 16`
`144 = 12 * 12`

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

Answer 4

15

Explanation:

there is another for you to observe @@
`144=2^4*3^2`
5 possibilities to choose `2^(0~4)`
3 possibilities to choose `3^(0~2)`
so
`(4+1)*(2+1)=15`

Answer 5

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 * 3^2`

1. Add `1` to each exponent:

= `2^(4+1) * 3^(2+1)`
= `2^5 * 3^3`

1. Find the products of the exponents:

= `5 * 3`
= `15`

Best of luck!