How many factors does 36^2 have?
2
8
24
25
26
25
Can someone please explain how to work this out?
How many factors does 36^2 have?
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And because of the rule jangojess cites above, perfect squares always have an *odd* number of divisors, because when you prime factorize a perfect square, the exponents are all even. When you add one to each power and multiply, you'll be multiplying odd numbers only, therefore getting an odd result. If you know that, you'll know without factorizing anything that the answer to the above question must be odd, and there's only one odd number among the answer choices.
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This is really cool Ian, thanks! So when I see something like:Ian Stewart wrote:And because of the rule jangojess cites above, perfect squares always have an *odd* number of divisors, because when you prime factorize a perfect square, the exponents are all even. When you add one to each power and multiply, you'll be multiplying odd numbers only, therefore getting an odd result. If you know that, you'll know without factorizing anything that the answer to the above question must be odd, and there's only one odd number among the answer choices.
How many factors does 81^99 have? I know that 81 is a perfect square and I know that an ODD number to the power of any positive integer is ODD, then the answer must be ODD. Since there was only one odd answer in the answer list, it looks like GMAC was doing this on purpose.
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rather than explaining the rule with variables - which will get tedious, not to mention difficult to understand - i'll give an illustration with specific numbers.Mozartain wrote:Hi Ian
could you please explain the rule Jangojess has given?
let's say you want the total number of factors of the number 432, whose prime factorization is (2^4)(3^3).
here's the way to think about it: you have to choose (a) how many 2's, and, separately, (b) how many 3's are in the factorization. then you'll multiply these numbers of possibilities, in the same way as you'd multiply the numbers of possibilities for any other independent decision (such as 3 shirts x 4 pairs of pants = 12 outfits).
* because the factorization contains 2^4, there are 5 ways to choose the number of 2's: none, one, two, three, or four. note that this is 4 + 1 possibilities (= exponent + 1).
* because the factorization contains 3^3, there are 4 ways to choose the number of 3's: none, one, two, or three. note that this is 3 + 1 possibilities (= exponent + 1).
* then you'd multiply 5 x 4 = 20 possibilities total.
the same approach works in general, for any exponents that may happen to appear in a prime factorization. this is the genesis of the formula cited above.
of course, because TIME CONTRAINTS are a big deal on this exam, you should also just memorize the formula (i.e., you shouldn't have to derive it from scratch when you take the test, because that's a tremendous waste of time).
Ron has been teaching various standardized tests for 20 years.
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How many factors does 36^2 have?
a) 2
b) 8
c) 24
d) 25
e) 26
Step 1:
Recognize this question is asking for all the numbers that go into the big number. Breakdown the big number into it's prime factorization - put into bases of 2, 3, 5, 7, etc...and assign exponents
Step 2
36^2 =
36 * 36
= 6 * 6 * 6 * 6
= 6^4
= (2*3)^4
= 2^4 * 3^4
Step 3:
Once we have the prime factorization (put into bases of 2, 3, 5, 7, etc), we can figure out the number of factors.
It turns out that the exponents can tell us how many factors are in that number. On a smaller scale, if we had the number 8, its factors are 1, 2, 4, 8.
8 = 2^3
All lower exponents of 2^3 are also factors, such as 2^2, 2^1 and 2^0. Thus 2^3 has 3 factors + "1" which is also a factor so 2^3 has 4 factors. In other words, you just take the exponent and add 1. Pretty simple, right?
Now that was for 8, what about if we multiplied by a different base like 3? so 8*3 = 24, which breaks down to 2^3 * 3. How many factors does 24 have?
Well 24 = 2* 2* 2* 3; the factors are basically any combination of the prime numbers. By adding the 3, we added more ways to combine the prime numbers. Now you can add all previous factors by 3 and create a new set of factors: 2^0 * 3, 2^1 *3, 2^2 * 3, 2^3 * 3 = {3, 6, 12, 24}
in addition to {1, 2, 4, 8}; essentially multiplying the number of factors by 2. 4*2 = 8 factors now
So this is what it looks like mathematically
24 = (2^3)*(3^1)
= (set of 4 factors) * (set of 2 factors) --we multiply because we want to find the number of combinations of 2's and 3's that form new factors
= 8 factors
Step 4)
Now, going back to our big number:
36^2 = 2^4 * 3^4
The number of factors are combinations of factors of 2^4 with factors of 3^4. So, the # of factors for 2^4 multipled by # of factors of 3^4 will give you the answer
2^4 = 16; has factors 2^0, 2^1, 2^2, 2^3, 2^4 = 1, 2, 4, 8, 16 (or 5 factors)
3^4 = 81; has factors 3^0, 3^1, 3^2, 3^3, 3^4 = 1, 3, 9, 27, 81 (or 5 factors)
Now we can combine any of the first set of factors with the second set of factors = 5*5 = 25
You can take {1, 3, 9, 27, 81} and multiply by 1.
Or you can multiply by 2. {3, 6, 18, 54, 162} are all factors of 36^2.
Or you can multiply by 4. By 8, By 16.
The largest number you get is 16*81 = 1296. And you will see this is equal to 36^2, so you know as a check that these two numbers multiplied still give you a factor. That factor being 1296, does go into 36^2.
Total there are 5*5 = 25 combinations of different factors generated.
The statement says: How many factors does 36^2 have?
So, it's asking for the total number of factors, aren't we me missing the negative facors?
So, it would be 25*2 = 50 factors total.
Isn't 25 the number of positive factors?
I'm kind of lost. Thanks for clarifying.
So, it's asking for the total number of factors, aren't we me missing the negative facors?
So, it would be 25*2 = 50 factors total.
Isn't 25 the number of positive factors?
I'm kind of lost. Thanks for clarifying.
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Good question. Normally, GMAT questions will restrict the values to consider only positive factors. In general, however, the GMAT allows for negative factors (divisors), as seen in this excerpt from the Official Guide:maxm wrote:The statement says: How many factors does 36^2 have?
So, it's asking for the total number of factors, aren't we me missing the negative facors?
So, it would be 25*2 = 50 factors total.
Isn't 25 the number of positive factors?
I'm kind of lost. Thanks for clarifying.
An integer is any number in the set {. . . -3, -2, -1, 0, 1, 2, 3, . . .}. If x and y are integers and x " 0, then x is a divisor (factor) of y provided that y = xn for some integer n. In this case, y is also said to be divisible by x or to be a multiple of x. For example, 7 is a divisor or factor of 28 since 28 = (7)(4), but 8 is not a divisor of 28 since there is no integer n such that 28 = 8n.
Since the original question does not restrict the factors to only positive factors, the correct answer here is 50.
Cheers,
Brent