Difference between revisions of "2001 AMC 10 Problems/Problem 6"
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| + | {{duplicate|[[2001 AMC 12 Problems|2001 AMC 12 #2]] and [[2001 AMC 10 Problems|2001 AMC 10 #6]]}} | ||
| + | |||
== Problem == | == Problem == | ||
| + | Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits | ||
| + | of the integer <math>n</math>. For example, <math>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a | ||
| + | two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit of <math>N</math>? | ||
| + | |||
| + | <math>\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9</math> | ||
| + | |||
| + | == Solution == | ||
| + | Denote <math>a</math> and <math>b</math> as the tens and units digit of <math>N</math>, respectively. Then <math>N = 10a+b</math>. It follows that <math>10a+b=ab+a+b</math>, which implies that <math>9a=ab</math>. Since <math>a\neq0</math>, <math>b=9</math>. So the units digit of <math>N</math> is <math>\boxed{(\textbf{E})\ 9}</math>. | ||
| + | |||
| + | ==Video Solution by Daily Dose of Math== | ||
| + | |||
| + | https://youtu.be/t6AxKdlda7I?si=ZIZP5ZyA0JN-Bgf9 | ||
| − | + | ~Thesmartgreekmathdude | |
| − | + | == See Also == | |
| + | {{AMC10 box|year=2001|num-b=5|num-a=7}} | ||
| + | {{AMC12 box|year=2001|num-b=1|num-a=3}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 15:12, 15 July 2024
- The following problem is from both the 2001 AMC 12 #2 and 2001 AMC 10 #6, so both problems redirect to this page.
Problem
Let 
and 
denote the product and the sum, respectively, of the digits
of the integer 
. For example, 
and 
. Suppose 
is a
two-digit number such that 
. What is the units digit of ?
Solution
Denote 
and 
as the tens and units digit of , respectively. Then 
. It follows that 
, which implies that 
. Since 
, 
. So the units digit of is 
.
Video Solution by Daily Dose of Math
https://youtu.be/t6AxKdlda7I?si=ZIZP5ZyA0JN-Bgf9
~Thesmartgreekmathdude
See Also
| 2001 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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 | ||
| 2001 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 1 |
Followed by Problem 3 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
