Difference between revisions of "2013 AMC 10B Problems/Problem 18"
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If the number is of the form <math>\overline{2bcd},</math> then it must be one of the numbers <math>2000, 2001, \dots, 2012.</math> Checking all these numbers, we find that only <math>2002</math> has the given property. | If the number is of the form <math>\overline{2bcd},</math> then it must be one of the numbers <math>2000, 2001, \dots, 2012.</math> Checking all these numbers, we find that only <math>2002</math> has the given property. | ||
Therefore, the number of numbers with the property is <math>45 + 1 = \boxed{46}.</math> | Therefore, the number of numbers with the property is <math>45 + 1 = \boxed{46}.</math> | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/2jNuQEfo1Rc | ||
+ | |||
+ | ~savannahsolver | ||
== See also == | == See also == | ||
{{AMC10 box|year=2013|ab=B|num-b=17|num-a=19}} | {{AMC10 box|year=2013|ab=B|num-b=17|num-a=19}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 22:46, 1 February 2021
Contents
Problem
The number has the property that its units digit is the sum of its other digits, that is
. How many integers less than
but greater than
have this property?
Solution
We take cases on the thousands digit, which must be either or
:
If the number is of the form
where
are digits, then we must have
Since
we must have
By casework on the value of
, we find that there are
possible pairs
, and each pair uniquely determines the value of
, so we get
numbers with the given property.
If the number is of the form then it must be one of the numbers
Checking all these numbers, we find that only
has the given property.
Therefore, the number of numbers with the property is
Video Solution
~savannahsolver
See also
2013 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.