Prime factorisation of 8400- \(2^4*3*5^2*7\)
Number of factors of 8400 =
5*2*3*2=60

Number of factors of 8400 which are not multiple of 3
5*2*3=30

Number of factors of 8400 which are multiple of 3
60-30 = 30

Factors which are multiple of 3 but less than 50-
3,6,12,15,21,24,30,42,48
Total-9

Number of factors of 8400 which are multiple of 3 but more than 50 are
30-9 = 21

Answer:- C

how do you get the number of factors of 8400 which are not multiple of 3? thanks in advance.

Let \(n=a^p*b^q*c^r*d^s\) be the prime factorisation of integer n,where a,b,c and d are prime factors of n and p,q,r and s are their powers respectively.
Then the number of factors of integer n is given by the formula \((p+1)(q+1)(r+1)(s+1)\)

In this question, to find the number of factors which are not multiple of 3, we will ignore 3 and consider the power of other prime factors to put in the above mentioned formula.

Prime factorisation of 8400
\(2^4*3*5^2*7\)

After ignoring 3, we have \(2^4*5^2*7\)
Number of factors which do not include 3 will be
(4+1)(2+1)(1+1) = 5*3*2 = 30

Here's a free video that explains kunal555's formula AND it explains why it works: http://www.gmatprepnow.com/module/gmat- ... /video/828

Cheers,
Brent
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how come you didn't include 18, 27, 33, 36, 39, and 45 in the above?

Are those greater than 50??

You wrote:

Factors which are multiple of 3 but less than 50-
3,6,12,15,21,24,30,42,48
Total-9

So I'm curious as to why you didn't include 18, 27, 33, 36, 39, and 45 in the list...

Hello,

Ignore my previous post as I thought you are asking something else.

But Look at the prime factorisation of 8400... The factors mentioned by you are not the factors of 8400.

For example... You can try divide 18 with 8400....

Did you get integer... NO

The same is true for all numbers mentioned by you.

Hope it helps!!

yes perfect, thank you!

Is there a shortcut to determine whether these nos are factors of 8400 or not ?

Prime Factorization of each multiple of 3 is the key to determine if the multiples of 3 are factors of 8400.

Prime Factorization of 8400 is 2^4 * 5^2 *3*7

For instance , prime factorization of 12 = 3 * 2^2 . Hence 12 is a factor of 8400
Prime factorization of 33 = 3 * 11 . Hence 33 is not a factor of 8400