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 \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.
The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE:
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 !