SOLUTION: Which of the following is divisible by 13? a) 12! + 1 b)15! - 1 c) 14! + 1 d) none of these

Algebra ->  Exponents -> SOLUTION: Which of the following is divisible by 13? a) 12! + 1 b)15! - 1 c) 14! + 1 d) none of these       Log On


   

Question 1039348: Which of the following is divisible by 13?
a) 12! + 1 b)15! - 1 c) 14! + 1 d) none of these

Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!


(a)   %2812%21%2B1%29%2F13=36846277, an integer

(b)   %2815%21-1%29%2F13=100590335999%2612%2F13, not an integer

(c)   %2814%21%2B1%29%2F13=6706022400%261%2F13, not an integer

Answer: (a) only 

Edwin

Answer by ikleyn(52852) About Me  (Show Source):
You can put this solution on YOUR website!
.
1. It was solved before in https://www.algebra.com/algebra/homework/divisibility/Divisibility_and_Prime_Numbers.faq.question.1038874.html


2. In order to find that 15! -1 is not divisible by 13, there is no need to calculate this number and then make division.

   Simply 15! contains 13 as a factor, therefore 15! is a multiple of 13.
   Hence, the previous number 15! - 1 is not a multiple of 13.

3. Same logic works for 14! + 1.

4. By answering these questions in https://www.algebra.com/algebra/homework/divisibility/Divisibility_and_Prime_Numbers.faq.question.1038874.html 
   I assumed that it it obvious.