N! is completely divisible by 13^(52). What is the sum of the digits

Question

N! is completely divisible by  13^{52} . What is the sum of the digits of the smallest such number N?

(A) 9

(B) 11

(C) 15

(D) 16

(E) 19

Are You Up For the Challenge: 700 Level Questions

Regor60 · Answer

Between 12 and 157 there are 12 multiples of 13. At 169 there are 2 multiples of 13.

13^14

Between 169 and 326 there are another 12 multiples of 13. At 338 there are 2 multiples of 13.

13^14 * 13^14 =13^28

Between 339 and 495 there again another 12 multiples of 13. At 507 there are 2 multiples of 13.

13^28*13^14 = 13^42

Need 10 more multiples of 13 to reach 13^52.

507+130=637

6+3+7=16,D

Posted from my mobile device

KarishmaB · Answer

N! = 1*2*3*....*12*13*14*.... *26*27*... 169*170*...
 
To be divisible by 52 13s, we must get 13 52 times in this product.

13 will give us the first 13 since 13 is prime and we cannot make it using any other smaller factors. 
26 will give second. 
39 will give third.
...
But 13*13 = 169 will give two 13s. 
13*14 will give us fifteenth 13.
...
Similarly, 13*26 will give two 13s 
13*27 will give us 29th 13.
...
and 13*39 will give two 13s too. 
13*40 will give us 43rd 13
...
so 13*49 will give us 52th 13.

Hence, if N = 13*49 = 637, then 637! will have 52 13s in it.

Sum of digits of 637 = 16

Answer (D)

versewriter · Answer

Here is the solution

We can assume that N! will be completely divided by 13^52. So the no. should be equal to or less than 13*52 i.e 676!

But we are forgetting that the question is asking the smallest no N. So we all know that 52 no also contains 26 and 49 which is also a multiple of 13. so this will gives us the extra 13, so our no. cannot be 676.

No 52 has total 03 multiple of 13 so can say that
13! has only one 13 (it will not extra 13, if we have to find the smallest N!)
26! has two 13 in no. 26 (it will give us the one extra 13 )
49! has a three 13 in no. 49 (it will give us two extra 13)

So these three extra 13 should be subtracted from no 52 to find the smallest N!, so our no. will be 49. Thus our result will be 49*13 = 637. 
So Option D will be our answer (6+3+7 = 16)

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

N! is completely divisible by 13^(52). What is the sum of the digits

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

N! is completely divisible by 13^(52). What is the sum of the digits

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards