For-any-number-x-gt-1-x-Z-x-can-be-expressed-as-a-combination-of-numbers-multiplied-together-e-g-10-5-2-20-5-4-5-2-2-100-10-10-5-2-5-2-x-p-1-e-1-p-2-e-2-p-n-e-n-where-p-n-i

Question Number 7289 by FilupSmith last updated on 21/Aug/16

For any number x > 1 : x \in Z

x can be expressed as a combination

of numbers multiplied together .

e . g .

10 = 5 \times 2

20 = 5 \times 4 = 5 \times 2 \times 2

100 = 10 \times 10 = 5 \times 2 \times 5 \times 2

∴ x = p_{1}^{e_{1}} p_{2}^{e_{2}} \dots p_{n}^{e_{n}} where p_{n} is the n th

prime factor

e_{n} is the exponent

e . g .

x = 810 = 8 \times 10 \times 11 = 4 \times 2 \times 5 \times 2 \times 11

x = 2^{4} \times 5^{1} \times 11 \Rightarrow p_{1} = 2, e_{1} = 4, e t c .

let ω (x) be the number of discrete prime factors

∴ ω (x) = n

x = \underset{t = 1}{\prod^{ω (x)}} p_{t}^{e_{t}}

Therefore the number of prime factors

is given by :

℧ (x) = \underset{t = 0}{\sum^{ω (x)}} e_{t}

Question

How many combinations are there to

represent x as the product of k numbers ?

(exluding 1 . e . g . 10 = 10 \times 1 \times 1 \times 1 \times 1)

e . g . 3 number \Rightarrow 30 = 3 \times 10 = 3 \times 5 \times 2

℧ (30) = 3! = 6 ways

My working so far :

x = a \times b \times \dots \times k combinations of k numbers

x = (a_{1} \times a_{2} \times \dots \times a_{ω (a)}) \dots (k_{1} \times \dots \times k_{ω (k)})

a_{1}, a_{2}, \dots, k_{1}, e t c are the prime factors

of a, b, \dots ., k

∴ any arramgement with the bracets work

so long as k brackets are present

let C_{k} (x) be the combination of ways

to express x as the product of k

integers (not equal to 1) .

x = (\underset{t = 1}{\prod^{ω (a)}} a_{t}^{e_{t}}) (\underset{t = 1}{\prod^{ω (b)}} b_{t}^{e_{t}}) \dots (\underset{t = 1}{\prod^{ω (k)}} k_{1}^{e_{t}})

C_{k} (x) = ? ? ?

Commented by FilupSmith last updated on 21/Aug/16

lets say you was x as the product of two integers .

i . e . k = 2

x = (a_{1} \times a_{2} \times a_{3} \dots a_{α}) (b_{1} \times b_{2} \times b_{3} \dots b_{β})

= (a_{1} \times b_{2} \times a_{3} \dots a_{α}) (b_{1} \times a_{2} \times b_{3} \dots b_{β})

e t c

x = (\underset{t = 1}{\prod^{ω (a)}} a_{t}^{α_{t}}) (\underset{k = 1}{\prod^{ω (b)}} b_{t}^{β_{t}})

C_{2} (x) = (\underset{t = 1}{\sum^{ω (a)}} α_{t} + \underset{t = 1}{\sum^{ω (b)}} β_{t})

C_{2} (x) = (℧ (a) + ℧ (b))!

therefore does :

C_{k} (x) = [℧ (a) + ℧ (b) + \dots + ℧ (k)]!

Commented by prakash jain last updated on 21/Aug/16

Just to make sure that I understand your

question :

You are looking for a formula for finding

in how many ways a number can be written

as a product of k integers . Correct ?

For this purpose do you count, for exmaple

2 \times 5 and 5 \times 2 as two different ways ?

Commented by FilupSmith last updated on 21/Aug/16

correct