Advanced Math/divisor

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Expert: - 4/23/2009


Question
1. Find the number of the positive integral divisors of 72^72.

2. Find the sum of the positive integral divisors of 72 excluding 1 and 72.

3. Find the sum of the greatest odd divisors of 1,2,3,4,...,32?


Answer

Largest Odd Factor
1. Is that 72 to the 72nd power?

There are 2 unique prime factors – 2 and 3.

The number 72 has 3-2’s and 2-3’s.
This means it has (3+1)(2+1) = 4*3 = 12 divisors.

The number 72² (5,194) has 6-2’s and 4-3’s.
This means it has (6+1)(4+1) = 7*5=35 divisors.
Note: only squares have an odd number of divisors.

The number 72^3 { 373,248 } has 9-2’s and 6-3’s.
That’s (9+1)(6+1) = 10*7 = 70 factors.
If you want a list, start with 2^0, 2^1, 2^2, …2^8, 2^9 as the first 10.
Every times this row is multiplied by 3, 10 more factors are gotten.
Since there are 6-3’s, we take the first 10 and add 100 to it 6 times,
Which gives 10*7 = 70 factors.
For example, the first 10 would be
1, 2, 4, 8, 15,32, 64, 128, 256, 512.
The next 10 would be
3, 6, 12, 24, 48, 96, 192, 384, 768, 1536.
The next 10 would be
9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608.
That’s 2-3’s.  If you want them all, do it 4 more times.

The number 72^72 { no, I have no clue, other than it’s over 10^100th )
has 216-2’s and 144-3’s as prime factors.
The total number of factors would be (216+1)(144+1) = 217*145 = 31,465.
To list them all out (what, you crazy?  I don’t think I’ll do it)
start with the first 216 factors 1, 2, 4, 8, 16, …., 2^214, 2^215, 2^216.
That is 217 numbers.
Multiply this entire row by 3 to get the next 217.
Continue to multiply the row be 3 repeatedly until it has been done 144 times.
That is all of the factors including 1 and the number itself.

Subtract off 2 is 1 and the number itself are to be thrown out.

2. It was found in 1 to be 4*3 –2 = 12 – 2 = 10.

3. Find the sum of the greatest odd divisors of the numbers between 1 and 32.
I have attached a file that has the factors.
Note that the first column is the numbers,
the second column is if 1 and the number are not allowed,
and the third column is if 1 and the number are allowed.

Scott A Wilson

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