SOLUTION: Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $12$, it has the same three digits, in reverse

Algebra ->  Decimal-numbers -> SOLUTION: Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $12$, it has the same three digits, in reverse      Log On


   

Question 1207584: Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $12$, it has the same three digits, in reverse order. What is $N$? (Express your answer in decimal.)
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


It appears that there is no solution.

My work is shown below; perhaps another tutor will find an error in my work and find a solution to the problem.

N = ABC base 12:
N=144A%2B12B%2BC

N = CBA base 7:
N=49C%2B7B%2BA

Set the two expressions for N equal to each other and solve the resulting equation for one variable in terms of the other two.

144A%2B12B%2BC=49C%2B7B%2BA
143A%2B5B=48C
5B=48C-143A
5B=%2845C-140A%29%2B%283C-3A%29
5B=%2845C-140A%29%2B3%28C-A%29
B=%289C-28A%29%2B3%28C-A%29%2F5

A, B, and C are positive integers less than 7, so in that last equation B, 9C, and 28A are all integers; that means 3%28C-A%29%2F5 must be an integer.

That means C-A must be a multiple of 5; and since C and A are positive integers less than 7, C=6 and A=1.

Then we have

B=9%286%29-28%281%29%2B3=29

But B has to be less than 6.

So there is no solution.

Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

I agree with tutor greenestamps.

To verify I wrote a quick python script to check numbers 100 through 666 in base 7.
The script couldn't find any solutions, but it did find these near-misses
300 base 7 = 103 base 12
301 base 7 = 104 base 12
which is a bit interesting.

I found a similar problem online dealing with bases 11 and 15.
The python script found that:
241 base 11 = 142 base 15
482 base 11 = 284 base 15

Furthermore,
204 base 7 = 402 base 5
102 base 7 = 201 base 5
and
361 base 7 = 163 base 11
502 base 7 = 205 base 11

There are likely many other pairs of values like this.
It's possible your teacher made a typo somewhere.