Difference between revisions of "2005 AMC 10A Problems/Problem 16"
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So <math>(10a+b)-(a+b)=9a</math> must have a units digit of <math>6</math> | So <math>(10a+b)-(a+b)=9a</math> must have a units digit of <math>6</math> | ||
| − | <math>a=4</math> is the only way this can be true. | + | This is only possible if <math>9a=36</math>, so <math>a=4</math> is the only way this can be true. |
So the numbers that have this property are <math>40</math>, <math>41</math>, <math>42</math>, <math>43</math>, <math>44</math>, <math>45</math>, <math>46</math>, <math>47</math>, <math>48</math>, <math>49</math>. | So the numbers that have this property are <math>40</math>, <math>41</math>, <math>42</math>, <math>43</math>, <math>44</math>, <math>45</math>, <math>46</math>, <math>47</math>, <math>48</math>, <math>49</math>. | ||
| − | Therefore the answer is <math>10\Rightarrow D</math> | + | Therefore the answer is <math>10\Rightarrow D</math> |
==See Also== | ==See Also== | ||
Revision as of 21:36, 21 May 2013
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
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 
