Question

the no of prime factors in (15)^17*(4)^9*(6)^8 ... with logic.

TIA

Answer 1

Fundamental theorem of arithmetic states that every composite number can be expressed as a product of two or more prime numbers.

Let N be a composite number and a,b & c are its prime factors. Then :

N = a^p * b^q * c^r

Following hold good :

Number of factors = (p+1)(q+1)(r+1)

Number of unique factors = 3

Number of prime factors = p+q+r

Sum of factors = (a^0+a^1+..+a^p)(b^0+b^1+..+b^q)(c^0+c^1+..+c^r)

Product of factors = N^(Number of factors/2)

Coming to the problem, factorize expression into its prime factors

4^11 x 7^5 x 11 = 2^22 x 7^5 x 11^1

Number of factors = (22+1)(5+1)(1 + 1) = 23 x 6 x 2 = 276

Number of unique factors = 3

Number of prime factors = 22 + 5+ 1 = 28