Problem Solving

In the number 36 :

1>How many total factors ?
2>How many Odd factors ?
3>How many even factors ?
4>How many prime factors ?

This is a self devised question, so no OA.

BTW IMO answers are
[spoiler]1>9
2>3
3>6
4>2[/spoiler]

Short cuts after expert opinions.

Anurag@Gurome,

I'm not familiar with your method of breaking down total factors. Can you explain in a bit more detail?

Thanks!

In the number 36 :

1>How many total factors ?
2>How many Odd factors ?
3>How many even factors ?
4>How many prime factors ?

This is a self devised question, so no OA.

BTW IMO answers are
[spoiler]1>9
2>3
3>6
4>2[/spoiler]

To determine the number of positive factors of an integer:

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

36 = 2^2 * 3^2. Adding 1 to each exponent and multiplying, we get (2+1)*(2+1) = 9 factors.

Here's the reasoning. To determine how many factors can be created from 36 = 2^2 * 3^2, we need to determine the number of choices we have of each prime factor:

For 2, we can use 2^0, 2^1, or 2^2, giving us 3 choices.
For 3, we can use 3^0, 3^1, or 3^2, giving us 3 choices.

Multiplying, we get 3*3 = 9 possible factors.

Another example: How many positive factors does 882 have?

882 = 2 * 3^2 * 7^2. Adding 1 to each exponent and multiplying, we get 2*3*3 = 18 factors.

To determine the number of odd positive factors of an integer:

1) Prime-factorize the integer
2) Add 1 to the exponent of each odd prime factor
3) Multiply

36 = 2^2 * 3^2. The only odd prime factor is 3, with an exponent of 2. Adding 1 to the exponent, we get 2+1 = 3 odd factors.

Number of even positive factors = Total possible factors - Odd factors = 9-3 = 6.

So, to calculate EVEN POSITIVE FACTORS, we must find the total factors and subtract ODD POSITIVE FACTORS from it?

From 36 = 2^2 * 3^2. Why can't we say that only EVEN prime factor is 2, with an exponent of 2 and adding 1 to the exponent, we get 2+1 = 3 EVEN factors ?

vk_vinayak wrote:So, to calculate EVEN POSITIVE FACTORS, we must find the total factors and subtract ODD POSITIVE FACTORS from it?

From 36 = 2^2 * 3^2. Why can't we say that only EVEN prime factor is 2, with an exponent of 2 and adding 1 to the exponent, we get 2+1 = 3 EVEN factors ?

This approach counts one combination that is NOT even (2�) but omits many combinations that ARE even (2*3, 2*3², etc.).
A factor will be EVEN if its prime-factorization includes AT LEAST ONE 2.
To directly count the EVEN positive factors of a positive integer, we could do the following:

1. Prime-factorize the integer.
2. Add 1 to every exponent OTHER THAN 2's exponent.
3. Multiply the results by 2's exponent.

To illustrate:
720 = 2� * 3² * 5¹
The total number of EVEN factors = (4)(2+1)(1+1) = 24.

The reason that we DON'T add 1 to 2's exponent is that an EVEN factor must include AT LEAST ONE 2, so 2� is not an option.
An even factor of 720 must include either 2¹, 2², 2³, or 2�.
Thus, the total number of options with regard to 2� is 4 -- the value of 2's exponent.