In how many ways can a number be written as a product of two different factors?

In how many ways can a number be written as a product of two different factors?

Let Number=x

Let Its prime factors are:

2^a*3^b*5^c…………….and so on

Number of factors

=(a+1)(b+1)(c+1)………… and so on

Required number of ways

=[(a+1)(b+1)(c+1)………… and so on]/2

If Required number of ways result to a.b then Required number of ways=a+1

Example 1:

In how many ways can 1500 be resolved into two factors?

(a)12

(b)18

(c)20

(d)10

Answer: 12

Explanation:

1500=2*750=2*2*3*5*5*5=2²5³3¹

Number of factors=(2+1)(3+1)(1+1)=3*4*2=24

Required number of ways=16/2=8

Example 2:

In how many ways can 1200 be resolved into two factors?

(a)12

(b)18

(c)20

(d)10

Answer: 12

Explanation:

1200=12*100=2*2*3*5*2*5*2=2⁴5²3¹

Number of factors=(4+1)(2+1)(1+1)=5*3*2=30

Required number of ways=30/2=15

Example 3:

In how many ways can the number 243 be resolved into two factors?

(a)12

(b)18

(c)20

(d)10

Answer: 12

Explanation:

243=3*3*3*3*3=3⁵

Number of factors=(5+1)=6

Required number of ways=6/2=3