M03-20

Question

If  n=15!+13 , which of the following is a divisor of  n ?

A. 15
B. 14
C. 13
D. 7
E. 2

Bunuel · Accepted Answer

Official Solution:

If  n=15!+13 , which of the following is a divisor of  n ?

A. 15
B. 14
C. 13
D. 7
E. 2

n=15! + 13 = 1*2*3*...12*13*14*15+13 . Since 13 is a factor of both  15!  and the additional 13 in the expression, we can conclude that 13 is a divisor of  n . In other words, we can factor out 13 from the expression to get  n=13*(1*2*3*...12*14*15+1) , hence 13 is definitely a divisor of  n .

Answer: C

sytabish · Answer

Sum or difference of multiples of a number will also be multiple of the number. 
Since, 15! and 13 are both multiples of 13, the result will also be a multiple of 13.
Thanks!

piyuusshh · Answer

I used the concept of trailing zeroes as the last digits are ....0013 we can see that it is not divisible by any no. other than it self.
I am doubtfull whether it was mere luck in the question or is this way of doing it is also correct?

Bunuel · Answer

15! + 13 = ...013 (actually it's 1,307,674,368,013)

Since the last digit is 3 (odd) we can get that it's not divisible by any even number (eliminate B and E) and not divisible by 5 because it does not end with 0 or 5 (eliminate A). But a number ending with 13 may or may not be divisible by 7 and may or may not be divisible by 13. For example, 413 is divisible by 7 and not divisible by 13.

banerjr1 · Answer

I solved it in the following method. It might be slightly more involved than Bunuel's solution.

n=15!+13 
Let  x  be the divisor.

\frac{n}{x} = \frac{(15!+13)}{x} 
 \frac{n}{x} = \frac{15!}{x}+ \frac{13}{x} 
The only value where  \frac{n}{x}  is an integer is where  x = 13 .

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Sarita24 · Answer

I like the solution - it’s helpful.

Vishnupriyasundar · Answer

I like the solution - it’s helpful.

DeathpoolXiX · Answer

I like the solution - it’s helpful.

rifatbutex42 · Answer

I like the solution - it’s helpful.

Fisayofalana · Answer

I like the solution - it’s helpful.

facilisomnis · Answer

I did not quite understand the solution. How would I know that 15! is divisible by 13??

Bunuel · Answer

Because 15! means the product of all integers from 1 to 15, inclusive:

15! = 1 * 2 * 3 * ... * 12 * 13 * 14 * 15

Since 13 is one of the numbers in this product, 15! automatically includes 13 as a factor. That makes 15! divisible by 13.

In general, n! is always divisible by every integer k where 1 ≤ k ≤ n.

vitaevitae · Answer

I like the solution - it’s helpful.

LukeOstertag · Answer

I like the solution - it’s helpful.

Priyanshu7297 · Answer

I did not quite understand the solution. how exactly do you actually know that 13 is the factor of 15!

Bunuel · Answer

Explained here: https://gmatclub.com/forum/m03-183605.html#p3631735 Hope it helps.

M03-20

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