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Hello everyone. So in 
this example, we're going to 

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count the number of even
divisors of the number 1280.

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Now when you see the
word “divisors”,

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you should be thinking maybe 
using divisors from prime factorization.

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so DFPF.

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But in order to do that, you
need to prime factor 1280.

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So first, we factor 1280.

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Well so here we have 1280, 

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now I notice immediately
that 4 is a factor.

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Right? So 4 goes into 12 three times.

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No remainder. Then 4 
goes into 8 twice,

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and 4 goes into 0 zero times.

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So if we factor out the 4
we have 4 times 320. 

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320 is easy to see, 
that's 32 times 10. 

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and all of these numbers are small 
enough that I can prime factor them.

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2 squared, times 2 to the 5,
times 2, times 5,

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that's 2 to
the 8 times 5.

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Now DFPF, so divisors
from prime factorization,

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says that the divisors of 
this number of the form,  

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“2 to the a times
 5 to the b”

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with a being a number
between 0 and 8, 

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and b being a number 
between 0 and 1.

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This gives us all
the divisors of 1280.

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However, to get just the even factors,
we need at least one copy of 2.

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And hence, we will increase 
the lower bound on a to 1,

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and thus, 1 is less than or equal to a 
which is less than or equal to 8.

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So, what are the possible
values now for a?

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Well a is now between 1 and 8, and so
thus, we have 7 plus 1 possible factors

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(the plus 1 is because we 
count both endpoints, so 

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we take 8 minus 1 that's 7
plus 1 for both endpoints),

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and that gives us the total number of
possibilities for the power of 2.

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And we're going to take 1
plus 1 which is equal to 2,

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and that's the total possibilities
for the factors of 5, right?

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It's either 5 to the 
0, or 5 to the 1.

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It's 1 plus 1, again the difference 
plus 1 for the fact that we count both
endpoints.

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That's 8 times 2 and that equals 
16 possible even divisors of 1280. 

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So you might want to
practice writing

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down the 16 possibilities 
just to get some extra

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divisor from prime 
factorization help.

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So thank you very much, and 
hopefully this gives you an idea of 

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how to use DFPF to compute
the divisors of a number.

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