How many factors does 36^2 have?Finding the Number of Factors of an Integer
Please help as to how to solve this problem with 1 minute !!
First make prime factorization of an integer n=a^p*b^q*c^r
, where a
, and c
are prime factors of n
, and r
are their powers.
The number of factors of n
will be expressed by the formula (p+1)(q+1)(r+1)
this will include 1 and n itself.Example:
Finding the number of all factors of 450: 450=2^1*3^2*5^2
Total number of factors of 450 including 1 and 450 itself is (1+1)*(2+1)*(2+1)=2*3*3=18
factors.Back to the original question:
How many factors does 36^2 have?36^2=(2^2*3^2)^2=2^4*3^4
--> # of factors (4+1)*(4+1)=25
Or another way: 36^2 is a perfect square, # of factors of perfect square is always odd (as perfect square has even powers of its primes and when adding 1 to each and multiplying them as in above formula you'll get the multiplication of odd numbers which is odd). Only odd answer in answer choices is 25.
Hope it helps.
Thanks a ton !!.. loved the approach !