In the Arithmetic Session 2, while finding the even factors of 4500 we did the following:

First, I did PF

= 2^2 * 5^3 * 3^2

Total of 36 factors

Find # of odd factor

= 3 * 4 = 12

Even factors = 36 - 12 = (24)

2^2 = 2

2^2 * 3^2 = 3 * 3 = 9

2^2 * 5^3 = 3 * 4 = 12

2^2 * 5^3 * 3^2 = 1

This adds up to 24 factors which is correct.

But now in the case of 180

= 3^2 * 2^2 * 5

Total of 18 factors

Odd # of factors

= 3 * 2 = 6

Even

2^2 = 2

2^2*3^2 = 9*

2^25^1 = 6

2^2*5^1*3^2 = 1

= 18 factors

Which now does not holdup.

Can anyone explain when this strategy works ? Or can identify mistake in my work

I would just approach it this way: when you find the ODD factors, you know there **is not** a two (just need to consider the 2^0 case). Similarly, to find the EVEN factors, you know that there **is** at least one two (just need to consider the cases of 2^1, 2^2, and all cases where the exponent of 2 > 0).

After prime factorization, and ONLY for the 2, instead of adding 1 to the exponent for the multiplication, just use the exponent of 2.

So to find the **even** factors of 4,500:

2^2 * 3^2 * 5^3 (prime factorization)

2 * 3 * 4 (exponents + 1, except for the two)

= 24 **even** factors.

To find the **even** factors of 180:

2^2 * 3^2 * 5^1 (prime factorization)

2 * 3 * 2 (exponents + 1, except for the two)

= 12 **even** factors

Can you explain how you are adding up the even factors? It is not clear to me what this section means. For each row, I was unable to identify a consistent rule as to what you are multiplying to get 2, 9, 12, and 1 factor:

- 2^2 = 2
- 2^2 * 3^2 = 3 * 3 = 9
- 2^2 * 5^3 = 3 * 4 = 12
- 2^2 * 5^3 * 3^2 = 1

I think this approach is too complex, but if you insist on breaking out into these scenarios, they would add up to 12:

- 2^2 = Exponent of 2 > 0, Exponents of 3 & 5 = 0

2 factors
- 2^2 * 3^2 = Exponents of 2 & 3 > 0, Exponent of 5 = 0

4 factors (2*2)
- 2^2 * 5^3 = Exponents of 2 & 5 > 0, Exponent of 3 = 0

6 factors (2*3)
- 2^2 * 5^3 * 3^2 = Exponents of 2 & 3 & 5 > 0

12 factors (2*3*2)