Problem Solving

How many factors does 36^2 have?
a.2
b.8
c.24
d.25
e.26
OA:D[spoiler][/spoiler]

shikh wrote:How many factors does 36^2 have?
a.2
b.8
c.24
d.25
e.26
OA:D[spoiler][/spoiler]

To determine the number of positive factors of an integer:

1) Prime-factorize the integer
2) Add 1 to each exponent
3) Multiply

36² = 2^4 * 3^4.
Adding 1 to each exponent and multiplying, we get:
(4+1)*(4+1) = 25 factors.

The correct answer is D.

Here's the reasoning:

To determine how many factors can be created from 36² = 2^4 * 3^4, we need to determine the number of choices we have of each prime factor:
For 2, we can use 2^0, 2^1, 2², 2³, or 2^4, giving us 5 choices.
For 3, we can use 3^0, 3^1, 3², 3³, or 3^4, giving us 5 choices.

To combine the number of choices we have of each prime factor, we multiply:
5*5 = 25 factors.

If a number can be represented in the form (a^m) * (b^n) * (c^o) * .. where a,b,c.. are distinct prime numbers and m,n,o..are non negative integers, then the total number of factors = (m+1)*(n+1)*(o+1)*..enough gyan, let us solve the problem.

How many factors does 36^2 have?

36^2 = 6^4 = (2*3)^4 = 2^4 * 3^4.
Now, we know that 2 and 3 are primes and 4 is a non negative integer.
The total number of factors = (4+1)*(4+1) = 5*5 = 25