Difference between revisions of "2021 Fall AMC 10A Problems/Problem 8"
(→Solution 2) |
|||
Line 30: | Line 30: | ||
~Steven Chen (www.professorchenedu.com) | ~Steven Chen (www.professorchenedu.com) | ||
+ | |||
+ | == Solution 3 == | ||
+ | If the tens digit is <math>a</math> and the ones digit is <math>b</math> then the number is <math>10+b</math> so we have the equation <math>10a + b = a + b^2</math>. We can guess and check after narrowing the possible cuddly numbers down to <math>13,14,24,25,35,36,46,47,57,68,78,89,</math> and <math>99</math>. (We can narrow it down to these by just thinking about how <math>a</math>'s value affects <math>b</math>'s value and then check all the possiblities.) Checking all of these we get that there is only <math>\boxed{1}</math> 2-digit cuddly number, and it is <math>89</math>. | ||
+ | |||
+ | ~Andlind | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2021 Fall|ab=A|num-b=7|num-a=9}} | {{AMC10 box|year=2021 Fall|ab=A|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 23:55, 25 November 2021
Problem
A two-digit positive integer is said to be if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
Solution 1
Note that the number By the problem statement, From this we see that must be divisible by This only happens when Then, Thus, there is only cuddly number, which is
~NH14
Solution 2
Denote this number as .
Hence, we have .
This can be written as .
Hence, . This implies .
Hence, either or . Because , or 1.
For , we get . This is a solution.
For , we get . However, recall that . Hence, this is not a solution.
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 3
If the tens digit is and the ones digit is then the number is so we have the equation . We can guess and check after narrowing the possible cuddly numbers down to and . (We can narrow it down to these by just thinking about how 's value affects 's value and then check all the possiblities.) Checking all of these we get that there is only 2-digit cuddly number, and it is .
~Andlind
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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.