1995 AIME Problems/Problem 10
Problem
What is the largest positive integer that is not the sum of a positive integral multiple of and a positive composite integer?
Solution
The requested number must be a prime number. Also, every number that is a multiple of
greater than that prime number must also be prime, except for the requested number itself. So we make a table, listing all the primes up to
and the numbers that are multiples of
greater than them, until they reach a composite number.
is the greatest number in the list, so it is the answer. Note that considering
would have shortened the search, since
, and so within
numbers at least one must be divisible by
.
Afterword
We are basically looking for a number where when a multiple of 42 is subtracted from it, the result is a prime number. In addition, any number that ends in a 5 is not prime (except for 5 itself). Since 42 keeps the parity the same and advances odd numbers in the unit digit, then we can conclude that the sought number is . To be exact,
.
-jackshi2006
Second Solution
Let our answer be . Write
, where
are positive integers and
. Then note that
are all primes.
If is
, then
because
is the only prime divisible by
. We get
as our largest possibility in this case.
If is
, then
is divisible by
and thus
. Thus,
.
If is
, then
is divisible by
and thus
. Thus,
.
If is
, then
is divisible by
and thus
. Thus,
.
If is
, then
is divisible by
and thus
. Thus,
.
Our answer is .
See also
1995 AIME (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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