Write a complete mathematical definition of (a,b), the greatest common divisor of a,b in \mathbb{Z}
1 answer:
Answer:
The greatest common divisor of two integers a and b (not both 0) is the largest integer that divides both a and b.
Step-by-step explanation:
Think for example of the numbers a=5, and b= -10. The greatest common divisor of 5 and -10, is the largest integer that divides both 5 and -10. We can find it by inspection (although there are more advanced methods to find it). We can list all integers that divide both 5 and -10.
-5 divides 5, and it also divides -10
-1 divides 5, and it also divides -10
1 divides 5, and it also divides -10
5 divides 5, and it also dividies -10
The LARGEST of them all is then 5, so 5 is the greatest common divisor of 5 and -10. The usual way to write it is
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Answer:
see the attached table
Step-by-step explanation:
We assume the formula of interest is ...
residual = (initial amount)(1 -k)^t . . . . . . assuming k is a positive number
Where T is the half-life, this formula can also be expressed as ...
residual = (initial amount)(1/2)^(t/T)
Then the relationship between k and T is ...
(1 -k)^t = (1/2)^(t/T)
or ...
1 -k = (1/2)^(1/T)
This lets us write k in terms of T as ...
k = 1 -(1/2)^(1/T)
and it lets us write T in terms of k as ...
log(1-k) = (1/T)log(1/2)
T = log(1/2)/log(1-k)
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The attached spreadsheet table implements these formulas to compute T from k and vice versa. Formatting is in % and to four decimal places as required by the problem statement.
