Difference between revisions of "2005 AMC 10A Problems/Problem 16"
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<math> \textbf{(A) } 5\qquad \textbf{(B) } 7\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 19 </math> | <math> \textbf{(A) } 5\qquad \textbf{(B) } 7\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 19 </math> | ||
| − | ==Solution== | + | ==Solution 1== |
Let the number be <math>10a+b</math> where <math>a</math> and <math>b</math> are the tens and units digits of the number. | Let the number be <math>10a+b</math> where <math>a</math> and <math>b</math> are the tens and units digits of the number. | ||
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Therefore the answer is <math>\boxed{\textbf{(D) }10}</math> | Therefore the answer is <math>\boxed{\textbf{(D) }10}</math> | ||
| + | |||
| + | ==Solution 2== | ||
| + | Let a two-digit number equal <math>10a+b</math>, where <math>a</math> and <math>b</math> are the tens and units digits of the number. | ||
| + | |||
| + | From the problem, we have <math>10a+b-(a+b)=9a</math> | ||
| + | |||
| + | Now let <math>9a=10x+y</math>, where <math>x</math> and <math>y</math> are the tens and units digits of the number. Then it must be that <math>y=6</math> since the units digit is <math>6</math> as stated in the problem. | ||
| + | |||
| + | Note that <math>10a</math> ends in <math>0</math>, but <math>9a</math> ends in <math>6</math>, so <math>a=4</math>. We need not to care about <math>b</math>, since it cancels out in the calculation. | ||
| + | |||
| + | So the answer is <math>\boxed{\textbf{(D) }10}</math>, since there are <math>10</math> numbers that have <math>a=4</math>. | ||
| + | |||
| + | ~BurpSuite | ||
==See Also== | ==See Also== | ||
Revision as of 02:59, 17 July 2023
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 
and 
are the tens and units digits of the number.
So 
must have a units digit of
This is only possible if 
, so 
is the only way this can be true.
So the numbers that have this property are 
.
Therefore the answer is 
Solution 2
Let a two-digit number equal , where and are the tens and units digits of the number.
From the problem, we have 
Now let 
, where 
and 
are the tens and units digits of the number. Then it must be that 
since the units digit is as stated in the problem.
Note that 
ends in 
, but 
ends in , so . We need not to care about , since it cancels out in the calculation.
So the answer is , since there are 
numbers that have .
~BurpSuite
See Also
| 2005 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
