Question

How do you find the prime factorization of 153?

Answer 1

`=3times3times17`

Explanation:

`153`
`=3times3times17`
Note that `17` is a prime number

Answer 2

prime factors of 153 = `{3,3,17}`

Explanation:

`underline(color(black)("Prime Factoring Algorithm"))`
[1] Let `n` be the positive integer to be factored
[2] Let `f` be `2` (the smallest prime)
[3] Let `L` be an empty list which will eventually contain the prime factors

[4] while `f^2 < n`

[5]`color(white)("XXX")`if `ndivf` is an integer
[6]`color(white)("XXXXXX")`add `f` to the list `L`
[7]`color(white)("XXXXXX")`replace the value of `n` with `(ndivf)`
[8]`color(white)("XXX")`else
[9]`color(white)("XXXXXX")`replace the value of `f` with the next prime larger than `f`
[10]`color(white)("XXX")`end if

[11] end while

[12] add `n` to the list `L`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let's see how this works with the given positive integer `153`.

#{: (underline(color(black)(n)),underline(color(black)(f)),underline(color(black)(L)),underline("comment")), (153,2,{},), (153,3,{},), (51,3,{3},), (17,3,{3,3},), (17,5,,5^2>=17), (,,{3,3,17},) :}#

Answer 3

`153 = 3xx3xx 17`

Explanation:

The problem with numbers bigger than about 100, is that we often don't know their factors.

The "RULES for DIVISIBILITY" are very useful and prevent us from wasting time by dividing by numbers which are not factors. If we can find just one factor, the others are usually easy to find.

Does 153 `div 2?`

Only even numbers are divisible by 2. 153 is odd so it has no even number as a factor.

Does 153 `div 3?`

If the sum of the digits is divisible by 3, then 3 is a factor.
(1+5+3 = 9), so 3 is a factor.

Does 153 `div 5?`

153 does not end in a 5 or 0, so 5 is not a factor.

We only need to look for factors smaller than `sqrt153` because any factor pair will have one of the factors less than `sqrt153.`

#sqrt153 = 12.4

The only prime numbers left to check would be 7 and 11.

But we know that 3 is a factor, so let's start by dividing by 3.

`153 div 3 = 51" 5+1 = 6, so 51 does" div 3`
`51 div 3 = 17" 17 is prime"`

`:. 153 = 3xx3xx17`

Answer 4

`3^2xx17`

Explanation:

Starting point

If you add the digits of 153 you have 1+5+3 = 9

As 9 is divisible by 3 exactly then 153 is also divisible by 3 exactly

Build a factor tree:

From this you observe that `3xx51=153` and `3xx17=51`

as 3 and 17 are prime factors your answer is:

`3xx3xx17` written as `3^3xx17`

You can obtain a list of prime factors by doing an internet search.