Question

In how many ways can 1080 be expressed as a product of two factors?

Answer 1

Answer:(Step-by-step explanation:)

In how many ways can the number 1080 be expressed as a product of 3 integers?

How anonymous is DuckDuckGo?

1080 = 2^3 * 3^3* 5

Ordered triplets :-

Distribute 3 power of 2's

A + B + C = 3

5c2 = 10 ways

Distribute 3 power of 3's

A + B + C = 3

5C2 = 10 ways

Distribute 1 power of 5's

A + B + C = 1

3c2 = 3 ways

Total

10*10*3 = 300 ways

Unordered triplets:-

Now a*a*a = 1080 is not possible as it is nit a perfect cube and "a" has to be a intiger

a*a*b = 1080

Or a^2*b = 1080

Find the no. Of perfect square which are a factor of 1080= 2^3 * 3^3 *5

To find perfect square power should be even like 0,2,4,6,...

2 can take value of 0,2 = 2 value

3 can take value of 0,2 = 2 value

5 can take value of 0 = 1 value

Total perfect square =2*2*1 = 4

These 4 cases can be counted 3!/2! = 3 times each

So, the 4 cases which have been counted 3 times each, let's first count them 3 more times each

300+(4*3) = 312

Now, as everything has been counted 6 times, we get the unordered triplets by simply dividing all the cases by 6

312/6 = 52 unordered tripettriplets

Negative intiger case:-

Now, we need to consider the negative integers as well.

Let us take ‘a’, ‘b’, ‘c’ positive

In the 4 cases where two are identical, we can have

(a,a,b); (-a,-a,b); (-a, a, -b)

In the rest of the cases we will have

(a,b,c); (-a,-b,c); (-a, b, -c); (a, -b,-c)

So first let us multiply everything by 4

(52*4) ways

Now, as we have counted the 4 cases (where two are identical) one extra time, let us subtract each of them once

(52*4 -4)

= 4 (52-1)

= 4 (51)

= 204 total ways <--- final answer

Step-by-step explanation: