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Question 1204377: If 244n = 1022four Findn
Found 4 solutions by josgarithmetic, MathLover1, math_tutor2020, greenestamps:
Answer by josgarithmetic(39625)   (Show Source):
Answer by MathLover1(20850)  (Show Source):
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
It sounds like the equation is  , meaning we're looking for the base n that makes that equation true.
244 base n = 1022 base 4
2n^2 + 4n + 4 = 1*4^3 + 0*4^2 + 2*4^1 + 2*4^0
2n^2 + 4n + 4 = 74
2n^2 + 4n + 4-74 = 0
2n^2 + 4n - 70 = 0
2(n^2 + 2n - 35) = 0
2(n - 5)(n + 7) = 0
n-5 = 0 or n+7 = 0
n = 5 or n = -7
Ignore the negative solution. Base numbers must be positive integers.
Therefore,
244 base 5 = 1022 base 4
or we can write it like this
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As a check,
244 base 5 = 2*5^2 + 4*5^1 + 4*5^0 = 74 base 10
which matches with the conversion of 1022 base 4.
The answer is confirmed.
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
One of the other tutors understood the poorly posed problem correctly and showed a good formal solution.
If finding the answer quickly is important and a formal mathematical solution is not required, you can find the answer with a little bit of mental arithmetic.
The problem would not make any sense if n were not an integer, so do some estimating to find the probable answer and then verify that answer.
1022 base 4 is a bit more than 4^3=64; 244 base n is between 2n^2 and 3n^2. Some quick mental calculations show the probable answer is n=5.
Now confirm that answer.
1022 base 4 = 64+2(4)+2 = 74
244 base 5 = 2(25)+4(5)+4 = 74
ANSWER: n=5
NOTE: For a student familiar with numbers in bases other than base 10, here is another way to evaluate 244 base 5:
244 base 5 = (300-1) base 5 = 3(25)-1 = 75-1 = 74
That method works because 4 is the largest digit in base 5, so the next number after 244 in base 5 is 300.
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