2016 AMC 10B Problems/Problem 24
Problem
How many four-digit integers , with
, have the property that the three two-digit integers
form an increasing arithmetic sequence? One such number is
, where
,
,
, and
.
Solution 1
The numbers are and
. Note that only
can be zero, the numbers
,
, and
cannot start with a zero, and
.
To form the sequence, we need . This can be rearranged as
. Notice that since the left-hand side is a multiple of
, the right-hand side can only be
or
. (A value of
would contradict
.) Therefore we have two cases:
and
.
Case 1
If , then
, so
. This gives
.
If
, then
, so
. This gives
.
If
, then
, so
, giving
.
There is no solution for
.
Added together, this gives us
answers for Case 1.
Case 2
This means that the digits themselves are in an arithmetic sequence. This gives us answers,
Adding the two cases together, we find the answer to be
.
Solution 2
Looking at the answer options, all the numbers are pretty small so it is easy to make a list.
Counting all the cases we get our answer of which is
-srisainandan6
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
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