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.
Gmat loves factors
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36 = (2^2)*(3^2)bblast wrote:In the number 36 :
1>How many total factors ?
2>How many Odd factors ?
3>How many even factors ?
4>How many prime factors ?
1. Number of total factors = (Number of ways to select any number of 2's out of 2)*(Number of ways to select any number of 3's out of 2) = (2 + 1)*(2 + 1) = 9
2. Number of total odd factors = (Number of ways to select no 2)*(Number of ways to select any number of 3's out of 2) = 1*(2 + 1) = 3
3. Number of even factors = (Number of ways to select at least one 2 out of 2)*(Number of ways to select any number of 3's out of 2) = (2)*(2 + 1) = 6
4. Number of prime factors = 2
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[email protected],
I'm not familiar with your method of breaking down total factors. Can you explain in a bit more detail?
Thanks!
I'm not familiar with your method of breaking down total factors. Can you explain in a bit more detail?
Thanks!
 GMATGuruNY
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To determine the number of positive factors of an integer: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]
1) Primefactorize 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) Primefactorize 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 = 93 = 6.
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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 ?
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 ?
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If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to (a+1)(b+1)(c+1)...aleph777 wrote: I'm not familiar with your method of breaking down total factors. Can you explain in a bit more detail?
Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) = 5x4x2=40
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This approach counts one combination that is NOT even (2â�°) but omits many combinations that ARE even (2*3, 2*3Â², etc.).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 ?
A factor will be EVEN if its primefactorization includes AT LEAST ONE 2.
To directly count the EVEN positive factors of a positive integer, we could do the following:
1. Primefactorize the integer.
2. Add 1 to every exponent OTHER THAN 2's exponent.
3. Multiply the results by 2's exponent.
To illustrate:
720 = 2â�´ * 3Â² * 5Â¹
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 2â�° is not an option.
An even factor of 720 must include either 2Â¹, 2Â², 2Â³, or 2â�´.
Thus, the total number of options with regard to 2â�´ is 4  the value of 2's exponent.
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If you have the prime factorization of an even number, and it looks like this:
(2^k) * some odd primes
then the ratio of the number of even factors to the number of odd factors is k to 1.
So if you take a number like:
120 = (2^3)(3)(5)
then the ratio of even to odd divisors is 3 to 1, and so 3/4 of the factors of 120 will be even, and 1/4 of the factors of 120 will be odd.
As a consequence, every even number has at least as many even divisors as odd divisors, and any multiple of 2^2 = 4 has at least twice as many even divisors as odd divisors.
(2^k) * some odd primes
then the ratio of the number of even factors to the number of odd factors is k to 1.
So if you take a number like:
120 = (2^3)(3)(5)
then the ratio of even to odd divisors is 3 to 1, and so 3/4 of the factors of 120 will be even, and 1/4 of the factors of 120 will be odd.
As a consequence, every even number has at least as many even divisors as odd divisors, and any multiple of 2^2 = 4 has at least twice as many even divisors as odd divisors.
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com