Maths-
General
Easy
Question
If the (n + 1) numbers a, b, c, d,…….. be all different and each of them a prime number, then the number of different factors (other than 1) of am.b. c.d….. is
- (m + 1) 2n
- (m + 1) 2n – 1
- (m + 1) 2n + 1
- none of these.
The correct answer is: (m + 1) 2n – 1
There are n prime factors (b, c, d,….) other than a. First we find the total no. of factors not involving any power of a.
No. of prime factors in the factor of expression No. of ways
1nC1
2nC2
3nC3
……….
….…….
nnCn
\ total no. of ways
=nC1 + nC2 + …..+ nCn = 2n-1
Again consider the factors having the powers of a.
Factor of expression = ax 
product of y prime no.
where 1 
x m and 0 y x
\ there are m values of ax and nCy values of product of y prime nos.
\ total no. of factors having the powers of a
=m (nC0 + nC1 + ….. +nCn)= m.2n
Thus, the total no. of factors of the expression
= m.2n + 2n – 1 = (m + 1) 2n – 1.
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