Difference between revisions of "2001 AMC 12 Problems/Problem 2"
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== Solution == | == 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 | + | 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>. |
== See Also == | == See Also == | ||
{{AMC12 box|year=2001|num-b=1|num-a=3}} | {{AMC12 box|year=2001|num-b=1|num-a=3}} | ||
{{AMC10 box|year=2001|num-b=5|num-a=7}} | {{AMC10 box|year=2001|num-b=5|num-a=7}} | ||
+ | {{MAA Notice}} |
Latest revision as of 00:28, 30 December 2023
- 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
.
See Also
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 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.