Question 1152636: convert 123y=83 base 10
Found 2 solutions by Alan3354, greenestamps:
Answer by Alan3354(69443)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
 Solve for y: 123 (base y) = 83 base 10.
123 (base y) = y^2+2y+3
Informally, since y is almost certainly a whole number, just do some estimation. 83 is too close to 9^2=81 for y to be 9; so try y=8.
8^2+2(8)+3 = 64+16+3 = 83
ANSWER: y = 8
Formally, we have
 or 
While both y=-10 and y=8 are solutions to the equation, we want a positive number for the base, so we reject the solution y = -10.
And although it is overkill for this particular problem, you can find the answer by doing some logical analysis:
(1) y is at least 4, because 1, 2, and 3 are digits in base y.
(2) y is less than 10, because 123 (base 10) is obviously greater then 83.
(3) Since the last digit in base y is 3, 83-3 = 80 must be divisible by y. That means y is either 4, 5, or 8.
And trying 4 and 5 finds those are too small, so the answer is 8.
That kind of logical reasoning, while not needed for this particular problem, can be useful in similar, more difficult problems.
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