self Maths Puzzle

On planet Jupiterthe people use a certain number system to the base 'n' (n > 2), Jerk, a resident of the planet, one day received twice his daily wage because the digits of this wage, which was a 2 digit number, were reversed. If the value of 'n' is the least possible value there the decimal representation of the difference between Jerk's correct wage for the day is

Read Solution (Total 1)

Let the base of the number system be n and the two digit number be ab
given 2(ab)n = (ba)n
⇒ nb + a = 2na + 2b
⇒ a(2n - 1) = b(n - 2)
⇒ab=n−22n−1

Now the maximum possible value of a or b is (n - 1) (base n) for n = 3 we get
ab=15 = which is not possible as in base 3, 5 won't exist.
Similarly, for n = 4
but for n = 5
We get ab=39i.e.13
a = 1, b = 3 is the only number possible when n is the least possible.

The correct wage = (13)5=(8)10
The actual wage paid = (31)5=(16)10
So difference = 8
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self Other Question

A man bought some oranges at the rate of 3 oranges for one rupee and equal number of oranges at the rate of 2 oranges for one rupee. What is his profit, if he sells 2 oranges for one rupee. I take a four-digit number and subtract from it the sum of its digits. In the result I strike off one of the digits and the remaining three digits of the result are 2, 4 and 6 (not necessarily in that order). Find the digit struck off by me.