How many numbers that are not divisible

[Guest]

How many numbers that are not divisible by 6 divide evenly into 264,600?

(A) 9
(B) 36
(C) 51
(D) 63
(E) 72


Hi,

This is from man cjallenge problem set.

Could you kindly show me what approach should I follow for these kind of sums, although challenge problems are not very important from GMAT perspective I feel these kinds of sums do show up on the exams..

What I usually do is first I factorize the total into prime numbers. Could you kindly tell me what should be the next step for these kind of sums where we have to find the number of factors.. Also what is the answer for above sum ..I know the total factors is 144, what next? Thnx

[RA]

Is the answer 63 (D)?

[kevincan]

264,600?

Write as a product of prime numbers: 7^2 3^3 2^3 5^2

144 factors, 81 of which are not multiples of 6

[RA]

We can not get to 51 from the prime factors of 264,600 and therefore the answer should be (C)

[Guest]

264,600 /6 =44100
44100/6=7350
7350/6=1225 no more divisible by 6

1225 = 5 x 5 x 7 x 7 = 4 factors
So 3 x 3 =9 is the answer.

[Guest]

where is this question explained ?

[sumit]

i also get 9 using the method shown by Guest; is it correct? can someone fm MGMAT staff help?

[RonPurewal]

here's a cool fact:
you can find the total number of factors of any number by:
1) taking the exponents of the prime numbers in the prime factorization of the number
2) adding 1 to each of the exponents
3) multipliying the resulting numbers

the explanation behind this method is in the newest edition of our number properties strategy guide, but, in the final analysis, all that really matters (for gmat purposes) is that you know how to execute the method.

in this case, there are (2 + 1)(3 + 1)(3 + 1)(2 + 1) = 3 x 4 x 4 x 3 = 144 total factors.

if you want numbers that are NOT multiples of 6, there are two ways to do this.
here's the full details of one, and a sketch of the other.

--

(1) calculate directly: (factors that are not multiples of 3) + (factors that are not multiples of 2) - (factors that are multiples of neither 2 nor 3)
note that you have to subtract out the latter term because those numbers are double-counted in the first two terms.

the factors that are not multiples of 3 are factors of the remaining prime factorization once the 3's are taken out, viz., (7^2)(2^3)(5^2).
there are (2 + 1)(3 + 1)(2 + 1) = 3 x 4 x 3 = 36 such numbers.

the factors that are not multiples of 2 are factors of the remaining prime factorization once the 2's are taken out, viz., (7^2)(3^3)(5^2).
there are (2 + 1)(3 + 1)(2 + 1) = 3 x 4 x 3 = 36 such numbers.

the factors that are multiples of neither 2 nor 3 are factors of the remaining prime factorization once the 2's and the 3's are taken out, viz., (7^2)(5^2).
there are (2 + 1)(2 + 1) = 3 x 4 x 3 = 9 such numbers.

so, there are 36 + 36 - 9 = 63 such factors in total.

how does this square up with the solution given in the archive?

--

(2)
sketch of indirect calculation:
(total factors of 264600) - (factors that ARE divisible by 6)
there are 144 total factors of 264600.
the factors that ARE divisible by 6 must include at least one '2' and at least one '3'. if you know how to calculate this**, then you'll get (2 + 1)(3)(3)(2 + 1), in which you don't add 1 to the 3's anymore. that's 3 x 3 x 3 x 3, or 81.
therefore, there are 144 - 81 = 63 desired factors.

--

**if anyone is interested, i suppose i could post the rationale behind this.

[krishnan.anju1987]

Hi,

Sorry for posting a question over here after such a long time.
I understood the first explanation but was unable to follow the second. Could you please elaborate it further.

[tim]

how about a slight variation on Ron's explanation:

again, let's calculate the number of factors that ARE divisible by 6. the easiest way to do this is to divide 264600 by 6 to get 44100. now the factors of 264600 that are divisible by 6 are in a one-to-one correspondence with all the factors of 44100.

take the prime factorization of 44100: 2^2 3^2 5^2 7^2

add 1 to each exponent and multiply those new exponents: 3*3*3*3 = 81, so 81 of the 144 factors ARE multiples of 6.

subtract 81 from 144 to get 63 factors that are NOT multiples of 6.

[krishnan.anju1987]

Thanks for your immediate response but I think I am still confused about certain parts in your explanation. Specifically I don't understand why the number was divided by 6 in the first place and then after finding out the factors of the new number (which contained both 2 and 3) why were the total number of factors found and removed from 144?

[tim]

we are looking for factors of 264600 that are not multiples of 6. if you get why there are 144 total factors of 264600, all we have to do is find how many ARE multiples of 6 and subtract that. if a number n is a factor of 264600 that is a multiple of 6, that means n times 6k will be 264600, where k is some integer. so we have n6k = 264600, which translates directly to nk = 44100. now all we have to find is how many n's there are such that nk = 44100. by definition, this means we are looking for how many factors 44100 has..