Difference between revisions of "1973 Canadian MO Problems/Problem 3"
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Latest revision as of 20:34, 16 December 2011
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 |