Question 257279: Please help me solve this-Some whole numbers have the property that if you remove their units’ digit and place that digit at the front of the number the result is the same as the triple of the original number.
Find the second smallest such whole number!
Answer by drk(1908)   (Show Source):
You can put this solution on YOUR website! Lets start with case 1 - a two digit number
Let "ab" be the original 2 digit number.
According to the rule above, the new number is "ba".
Now ab can be expressed as 10A + B, where A is the tens digit and B is the units digit.
and ba can be expressed as 10A + B, where A is the tens digit and B is the units digit.
again, according to the rule above
distributing, we get
rearranging a bit, we get
and A/B is
there are no solutions for 2 digit numbers.
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Now case 2 - a three digit number
Let "abc" be the original 3 digit number.
According to the rule above, the new number is "cab".
Now abc can be expressed as 100A + 10B + C, where A is the hundreds digit, B is the tens digit, and C is the units digit
and cab can be expressed as 100C + 10A + B.
again, according to the rule above
distributing, we get
rearranging a bit, we get
Solving for C, we get
If A = 9, B = 7, and C = 29 - -> no solution
So, there are no solutions for 3 digit numbers.
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We could continue with 4 and five digit numbers, but that would be very complicated.
I have have found close solutions at
103; off by 1
206; off by 2
309; off by 3
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unfortunately I don't see the "key" that would get the answer here.
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