How many factors does 36^2 have?
a) 2
b) 8
c) 24
d) 25
e) 26
Step 1:
Recognize this question is asking for all the numbers that go into the big number. Breakdown the big number into it's prime factorization - put into bases of 2, 3, 5, 7, etc...and assign exponents
Step 2
36^2 =
36 * 36
= 6 * 6 * 6 * 6
= 6^4
= (2*3)^4
= 2^4 * 3^4
Step 3:
Once we have the prime factorization (put into bases of 2, 3, 5, 7, etc), we can figure out the number of factors.
It turns out that the exponents can tell us how many factors are in that number. On a smaller scale, if we had the number 8, its factors are 1, 2, 4, 8.
8 = 2^3
All lower exponents of 2^3 are also factors, such as 2^2, 2^1 and 2^0. Thus 2^3 has 3 factors + "1" which is also a factor so 2^3 has 4 factors. In other words, you just take the exponent and add 1. Pretty simple, right?
Now that was for 8, what about if we multiplied by a different base like 3? so 8*3 = 24, which breaks down to 2^3 * 3. How many factors does 24 have?
Well 24 = 2* 2* 2* 3; the factors are basically any combination of the prime numbers. By adding the 3, we added more ways to combine the prime numbers. Now you can add all previous factors by 3 and create a new set of factors: 2^0 * 3, 2^1 *3, 2^2 * 3, 2^3 * 3 = {3, 6, 12, 24}
in addition to {1, 2, 4, 8};
essentially multiplying the number of factors by 2. 4*2 = 8 factors now
So this is what it looks like mathematically
24 = (2^3)*(3^1)
= (set of 4 factors) * (set of 2 factors) --we multiply because we want to find the number of combinations of 2's and 3's that form new factors
= 8 factors
Step 4)
Now, going back to our big number:
36^2 = 2^4 * 3^4
The number of factors are combinations of factors of 2^4 with factors of 3^4. So, the # of factors for 2^4 multipled by # of factors of 3^4 will give you the answer
2^4 = 16; has factors 2^0, 2^1, 2^2, 2^3, 2^4 =
1, 2, 4, 8, 16 (or 5 factors)
3^4 = 81; has factors 3^0, 3^1, 3^2, 3^3, 3^4 =
1, 3, 9, 27, 81 (or 5 factors)
Now we can combine any of the first set of factors with the second set of factors = 5*5 = 25
You can take {1, 3, 9, 27, 81} and multiply by 1.
Or you can multiply by 2. {3, 6, 18, 54, 162} are all factors of 36^2.
Or you can multiply by 4. By 8, By 16.
The largest number you get is 16*81 = 1296. And you will see this is equal to 36^2, so you know as a check that these two numbers multiplied still give you a factor. That factor being 1296, does go into 36^2.
Total there are 5*5 = 25 combinations of different factors generated.
