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Question 1103484: A three digit number is equal to 17times sum of its digit. If 198 is added to the number the extreme digit get interchanged. The sum of the first and third digit is 1 less than the middle term. Find the number.
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! let a = the 100's digit
let b = the tens
let c = the units
:
Write an equation for each statement, simplify as much as you can
:
A three digit number is equal to 17 times sum of its digit.
100a + 10b + c = 17(a+b+c)
100a + 10b + c = 17a + 17b + 17c
100a - 17a + 10b - 17b + c - 17c = 0
83a - 7b - 16c = 0
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If 198 is added to the number the extreme digit get interchanged.
100a + 10b + c + 198 = 100c + 10b + a
100a - a + 10b - 10b +c - 100c = -198
99a - 99c = -198
simply divide by 99
a - c = -2
Rearrange
c = a + 2
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The sum of the first and third digit is 1 less than the middle term.
a + c = b - 1
a - b + c = -1
Using these two equations
a - b + c = -1
a + 0 - c = -2
----------------Adding eliminates c
2a - b + 0 = -3
2a - b = -3
Rearrange
b = 2a + 3
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Using the 1st simplified equation: 83a - 7b - 16c = 0
replace a with (2a+3); replace c with (a+2)
83a - 7(2a+3) - 16(a+2) = 0
83a - 14a - 21 - 16c - 32 = 0
83a - 30a - 53 = 0
53a = 53
a = 1
then
b = 2(1) + 3
b = 5
and
c = 1 + 2
c = 3
Find the number. 153 is the number
:
:
Check this in the first statement
"A three digit number is equal to 17 times sum of its digit.
153 = 17(1+5+3)
153 = 17(9)
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
A formal algebraic solution to this problem is a good exercise.
But there is a big shortcut you can take if you are just looking to get the answer as quickly as possible -- as, for example, if the question is on a competitive timed test.
The difference between a 3-digit number and the number with the same digits reversed is always a multiple of 99. Specifically, if the difference between the two numbers is 198 = 2*99, then the first and last digits of the original number differ by 2.
In this problem, 198=2*99 is added to the original number to get the new number; that means the possibilities for the original number are 1?3, 2?4, 3?5, 4?6, etc.
The additional given information that the middle digit is 1 more than the sum of the first and third digits limits the possible answers to 153, 274, and 395.
Finally the requirement that the original number must be equal to 17 times the sum of the digits shows that the original number has to be 153.
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