If a = 15! + 13, which of the following are factors of a?

Question

I. 13
II. 14
III. 26

A. I only
B. I and II only
C. I and III only
D. I, II, and III only
E. None of the above

Archit3110 · Answer

f a = 15! + 13, which of the following are factors of a?

I. 13
II. 14
III. 26

A. I only
B. I and II only
C. I and III only
D. I, II, and III only
E. None of the above

IMO only option A would be correct..
from the given expression a=15!+13
we can get : 13((15*14*12!)+1)
by which we can say that for factors of a would be 13 so option 1 is correct.

Coming to 2nd option 14: factor 7*2 so no

and 3rd option 26= 13*2, 
 for the expression 13((15*14*12!)+1) , though 13 would get cancelled out but for the rest of expression the outcome would be an odd no. so 26 wont be factor of the expression..

This is my take of the question, let me know if i have done any error..

GMATinsight · Answer

a = 15! + 13 = 13xEven+13 = 13(Even+1) = 13xOdd number

Hence, only 13 will be factor of a out of gievn three numbers

Answer: Option A

KarishmaB · Answer

a = 15! + 13 = 13 * (1*2*3..12*14*15 + 1)

Note the two factors of a - "13" and "1 more than a number that has all other primes till 15"
Hence, this second factor will have none of these primes (2, 3, 5, 7, 11)

I. 13
13 is a factor of a.

II. 14
2 and 7 are not factors of a.

III. 26
2 is not a factor of a.

Ved22 · Answer

Karishma in a simple way, we can also look at this problem. Because 15! will have 13 as a factor and N= 15!+13 so on the number line it will be a multiple of 13 + 13,, N+13 so the number will be only divisible by 13 and no other integer in options

KarishmaB · Answer

Yes, it is easy to say why the number will be a multiple of 13. What does need to be noted though is why the number will not be a multiple of 26 which is just 2*13. So we need to notice why 2 is not a factor of a. 
To do that we have multiple ways - Either notice that 15! will be even and if we add 13 to it, even + odd = odd so 2 will not be a factor. 
Or note that here 13* ( 1*2...*12*14*15  + 1), the underlined will have 2 as a factor so ( 1*2...*12*14*15  + 1) will not.

P.S. - Request you to tag me if you would like me to give some input. I am unable to see all the replies.

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If a = 15! + 13, which of the following are factors of a?

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If a = 15! + 13, which of the following are factors of a?

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