Difference between revisions of "1997 PMWC Problems/Problem T9"
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==Problem== | ==Problem== | ||
| − | Find the two 10-digit numbers which become nine times as large if the order of the digits is reversed. | + | Find the two <math>10</math>-digit numbers which become nine times as large if the order of the digits is reversed. |
==Solution== | ==Solution== | ||
| − | { | + | The pair of numbers are <math>1089001089</math> and is <math>1098910989</math>. |
| + | |||
| + | Notice that the sum of two numbers is 10 times of the smaller one. Let the smaller one be <math>a_9a_8a_7a_6a_5a_4a_3a_2a_1a_0</math>, the large one becomes <math>a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9</math>. Then we have | ||
| + | <math>a_9a_8a_7a_6a_5a_4a_3a_2a_1a_010</math> = <math>a_9a_8a_7a_6a_5a_4a_3a_2a_1a_0</math>+<math>a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9</math>. | ||
| + | It's obvious that <math>a_9=1</math> and <math>a_0=9</math>. Comparing the digits, we have <math>(a_8=0, a_1=8)</math>, <math>(a_7=8, a_2=0)</math>, <math>(a_6=9, a_3=1)</math>, and <math>(a_5=0, a_4=0)</math>. | ||
| + | |||
| + | |||
| + | Side note: | ||
| + | if its not clear how to obtain the digits | ||
| + | essentially we want to write an equation first and then compare the units,tens,hundreds digits etc.. on the left and right side. its easy to notice the digits must be one of {0,1,8,9} hence there aren't many cases to consider was well | ||
==See Also== | ==See Also== | ||
{{PMWC box|year=1997|num-b=T8|num-a=T10}} | {{PMWC box|year=1997|num-b=T8|num-a=T10}} | ||
| − | [[Category: | + | [[Category:Intermediate Number Theory Problems]] |
Latest revision as of 15:44, 6 August 2021
Problem
Find the two 
-digit numbers which become nine times as large if the order of the digits is reversed.
Solution
The pair of numbers are 
and is 
.
Notice that the sum of two numbers is 10 times of the smaller one. Let the smaller one be 
, the large one becomes 
. Then we have
= +.
It's obvious that 
and 
. Comparing the digits, we have 
, 
, 
, and 
.
Side note:
if its not clear how to obtain the digits
essentially we want to write an equation first and then compare the units,tens,hundreds digits etc.. on the left and right side. its easy to notice the digits must be one of {0,1,8,9} hence there aren't many cases to consider was well
See Also
| 1997 PMWC (Problems) | ||
| Preceded by Problem T8 |
Followed by Problem T10 | |
| I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
