In how many ways can a number be written as a product of two different factors?
Case 1: Non square no:
Let’s take a simple no: 84.
Factors of 84 are:
1,2,3,4,6,7,12,14,21,28,42,84.
Total: 12 Let’s check if we miss any factor:
Total no of factors for 84: [(2^2)*3*7] = 3*2*2=12.
So, we captured all the factors of 84.
Now 84 can be expressed as a product of 2 natural no’s:
1*84, 2*42, 3*28, 4*21, 6*14, 7*12 ------- 6 Ways So if a no has n factors, it can be expressed as a product of 2 natural no’s as:
---- {(First factor from left)*(first factor from right)} ,
{(second factor from left)*(second factor from right)} ,
till………………………. {([n/2]nd factor from left) * ([n/2]nd factor from right)}.
So, if a no has ‘N’ factors, where N is even (for non square number), total no of ways it can be expressed as product of 2 natural no’s: N/2.
If you are clear till this point, you are thinking about what if a no has odd no of factors.
This leads to our case no 2.
Case 2: Square number: As many of you might be aware that Only a square number has odd no of factors. Or if a no has odd no of factors, then it has to be a square no.
lets take a simple no: 36
Factors of 36: 1,2,3,4,6,9,12,18,36.
Total no of factors: 9.
36 can be expressed as a product of 2 different natural no’s:
1*36, 2*18, 3*12, 4*9, 6*6. ------ 5 ways.
If you notice, middle no is expressed as a product to itself (6*6).
So, for a square no having no of factors as N, total no of ways are:
(N+1)/2 ------ If repetition of numbers is allowed.
(N-1)/2------- If both the no’s has to be unique.
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