SOLUTION: If 321n=232seven find n

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Question 1195856: If 321n=232seven find n


Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
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Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The problem is this:

321 (base n) = 232 (base 7)

Informally....

In any base (where the numbers are defined), 321 is greater than 232; so if 321 (base n) is equal to 232 (base 7), then the base n must be less than 7.

So try n=6....

321 (base 6)
3*6=18; 18+2=20
20*6=120; 120+1=121

232 (base 7)
2*7=14; 14+3=17
17*7=119; 119+2=121

n=6 works!

ANSWER: n=6

Formally....

321 (base n) = 3n^2+2n+1
232 (base 7) = 2(7^2)+3(7)+2 = 98+21+2 = 121

3n^2+2n+1=121
3n^2+2n-120 = 0
(n-6)(3n+20) = 0

n=6 or n=-20/3

Obviously the negative fractional answer makes not sense. So

ANSWER: n=6