Conversion from decimal to unknown number system

Question

If we have the number $(387)_{10} \rightarrow (762)_n$ , how do we calculate the $n$? Thanks in advance.

Answer 1

HINT:

We have $$387=7n^2+6n+2$$ where integer $n>7$(why?)

Answer 2

387 in decimal that is 603 in octal and 762 is then base 8 and a fraction

n = 8+k 7n^2 + 6n + 2 = 387 7n^2 + 6n - 385 = 0 is a quadratic with standard form ax2 + bx + c = 0 and when we rearrange x = (-b +- (b^2-4ac)^0.5)/2a substituting x with n n = (-6 +-(6^2-4x7x-385)^0.5)/2x7 n = (-6 +- 104 )/14 that is n= 7 or -7.857142857142857

7x49+6x7+2= 432.14-47.14+2= 387

I now tried also for 0 in decimal and 767 base n see my post https://www.linkedin.com/pulse/proof-0-number-peter-paul-troendle

its really a nice question thanks I had a lot of fun thanks~