+0  
 
0
79
2
avatar

A 4 digit number is equal to the digit sum of the number multiplied by 109.Find the biggest numbber satisfied the above condition

Guest Aug 31, 2017
Sort: 

2+0 Answers

 #1
avatar+78575 
+1

 

 

Here's my best attempt at this one......whether it's the correct answer, I don't know.....!!!

 

Note that any four digit number can be written as

 

1000a + 100b + 10c + d

 

And let the sum of its digits be     a + b + c + d

 

And we're told that

 

1000a + 100b + 10c + d =  109 (a + b + c + d)     distribute the 109

 

1000a + 100b + 10c + d  = 109a + 109b + 109c + 109d       subtract the right side from both sides  

 

891a - 9b -99c - 108d  = 0          divide through by 9

 

99a - b - 11c - 12d = 0        add  12d , b to both sides

 

99a - 11c =  12d + b            factor the left side

 

11 [ 9a - c ]  = `12d + b         divide both sides by 11

 

9a - c  =       [ 12d + b ] / 11

 

Note that we need to have "a" as large as possible.......

 

And as large as the right side can be, 10, is when d = 9 and b = 2

And this implies that a = 2  and c =8 

 

And when the right side is 9, d = 8 and b = 3

And this implies that a = 2 and c = 9

 

And when the right side is 8, d = 7 and b = 4 

And this implies that  a =  2, but c = 10 which is impossible

 

So....it appears that the largest our number can be is  when a = 2, b = 3, c = 9 and d = 8

 

So.....the number is  2398 

 

P.S.  -  any corrections / additional answers by other mathematicians are welcome !!!!!!! 

 

 

cool cool cool

CPhill  Aug 31, 2017
edited by CPhill  Sep 1, 2017
 #2
avatar+18712 
+1

A 4 digit number is equal to the digit sum of the number multiplied by 109.

Find the biggest numbber satisfied the above condition

 

\(\begin{array}{|rccccl|} \hline 1. & 1 & 0 & 9 & 0 \\ 2. & 1 & 3 & 0 & 8 \\ 3. & 1 & 4 & 1 & 7 \\ 4. & 1 & 5 & 2 & 6 \\ 5. & 1 & 6 & 3 & 5 \\ 6. & 1 & 7 & 4 & 4 \\ 7. & 1 & 8 & 5 & 3 \\ 8. & 1 & 9 & 6 & 2 \\ 9. & 2 & 2 & 8 & 9 \\ 10. & \mathbf{2} & \mathbf{3} & \mathbf{9} & \mathbf{8} & \text{max} \\ \hline \end{array} \)

 

laugh

heureka  Sep 1, 2017

2 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details