2005 AMC 10A Problems/Problem 16
Contents
Problem
The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is . How many two-digit numbers have this property?
Solution 1
Let the number be , where
is its tens digit and
is its units digit. Then
must have a units digit of
, and as
is the tens digit, we can only have
, so
.
Accordingly, has units digit
only if
. Thus the numbers that have the required property are all those with tens digit
, from
to
, so the answer is
.
Solution 2
As in Solution 1, suppose that and
are the tens and units digits of the number respectively, so the result of the subtraction is
. Thus
must have units digit
.
We now observe that is a multiple of 10, so has units digit
, and hence
will have units digit
(where the
will become
in the subtraction by 'borrowing'
from the tens digit). Since
is a single digit, this simply means
, while
can be any digit from
to
(since it cancelled out in the subtraction above).
Thus, as in Solution 1, there are possible choices for
with
, so the answer is
.
~BurpSuite
See Also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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All AMC 10 Problems and Solutions |
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