1993 AIME Problems/Problem 6
Contents
Problem
The sum of 9 consecutive integers is 9. What is the least of these 9 integers?
Solution
Solution 1
Denote the first of each of the series of consecutive integers as . Therefore,
. Simplifying,
. The relationship between
suggests that
is divisible by
. Also,
, so
is divisible by
. We find that the least possible value of
, so the answer is
.
Solution 2
Let the desired integer be . From the information given, it can be determined that, for positive integers
:
This can be rewritten as the following congruences:
Since 9 and 11 are relatively prime, n is a multiple of 99. It can then easily be determined that the smallest multiple of 99 with a units digit 5 (this can be interpreted from the 2nd congruence) is
Solution 3
Let be the desired integer. From the given information, we have
here,
and
are the middle terms of the sequence of 9 and 11 numbers, respectively. Similarly, we have
as the 4th term of the sequence. Since,
is a multiple of
and
it is also a multiple of
Hence,
for some
So, we have
It follows that
is the smallest integer that can be represented in such a way.
Solution 4
By the method in Solution 1, we find that the number can be written as
for some integers
. From this, we can see that
must be divisible by 9, 5, and 11. This means
must be divisible by 495. The only multiples of 495 that are small enough to be AIME answers are 495 and 990. From the second of the three expressions above, we can see that
cannot be divisible by 10, so
must equal
. Solution by Zeroman.
See also
1993 AIME (Problems • Answer Key • Resources) | ||
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