Difference between revisions of "2020 AIME I Problems/Problem 10"
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Let <math>m</math> and <math>n</math> be positive integers satisfying the conditions | Let <math>m</math> and <math>n</math> be positive integers satisfying the conditions | ||
Revision as of 20:03, 19 June 2020
Problem
Let and
be positive integers satisfying the conditions
is a multiple of
and
is not a multiple of
Find the least possible value of
Solution 1
Taking inspiration from we are inspired to take
to be
, the lowest prime not dividing
, or
. Now, there are
factors of
, so
, and then
for
. Now,
. Noting
is the minimal that satisfies this, we get
. Thus, it is easy to verify this is minimal and we get
. ~awang11
Solution 2
Assume for the sake of contradiction that is a multiple of a single digit prime number, then
must also be a multiple of that single digit prime number to accommodate for
. However that means that
is divisible by that single digit prime number, which violates
, so contradiction.
is also not 1 because then
would be a multiple of it.
Thus, is a multiple of 11 and/or 13 and/or 17 and/or...
Assume for the sake of contradiction that has at most 1 power of 11, at most 1 power of 13...and so on...
Then, for
to be satisfied,
must contain at least the same prime factors that
has. This tells us that for the primes where
has one power of,
also has at least one power, and since this holds true for all the primes of
,
. Contradiction.
Thus needs more than one power of some prime.
The obvious smallest possible value of
now is
.
Since
, we need
to be a multiple of 11 at least
that is not divisible by
and most importantly,
.
is divisible by
, out.
is divisible by 2, out.
is divisible by 5, out.
is divisible by 2, out.
and satisfies all the conditions in the given problem, and the next case
will give us at least
, so we get
.
Video Solution
Video Solution
https://www.youtube.com/watch?v=FQSiQChGjpI&list=PLLCzevlMcsWN9y8YI4KNPZlhdsjPTlRrb&index=7 ~ MathEx
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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