1973 Canadian MO Problems/Problem 3
Problem
Prove that if 
and 
are prime integers greater than 
, then 
is a factor of 
.
Solution
Prime numbers greater than are odd. Thus, if and are prime integers greater than , then they are odd, and is a multiple of 
. Also, consider each group of three consecutive integers. One has remainder 
after division upon , one has remainder , and one has remainder 
. If and are prime integers greater than , then they cannot be divisible by . Thus, must leave remainder after division by three, and so is a multiple of .
Finally, if is a multiple of and , then it is a multiple of 
.
See also
| 1973 Canadian MO (Problems) | ||
| Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 | Followed by Problem 4 |
