The biggest remainder obtained by dividing 2015 by either 1, 2, 3,
4,...,1000 is the option;
(C) 671
How can the biggest remainder be found?
The given number Maria divides is 2015
The divisors = 1, 2, 3, 4,..., 1000
Required:
The biggest remainder Maria noted down.
Solution:
From remainder theorem, we have;
p(x) = (x - a) × q·(x) + r(x)
Where;
r(x) = The remainder
q(x) = The quotient
(x - a) = The devisor
Which gives;
[tex]\dfrac{r(x) }{(x - a)} = \dfrac{p(x) }{(x - a)} -\dfrac{(x - a) \times q(x) }{(x - a)} = \mathbf{ \dfrac{p(x) }{(x - a)} -q(x)}[/tex]
Therefore;
[tex]r(x) = \mathbf{\left(\dfrac{p(x) }{(x - a)} -q(x) \right) \times (x - a)}[/tex]
The remainder is largest when both [tex]\left(\dfrac{p(x) }{(x - a)} -q(x) \right)[/tex] and (x - a) are large
[tex]\left(\dfrac{p(x) }{(x - a)} -q(x) \right)[/tex] is largest when the quotient changes to the next lower
digit and the divisor is not a factor.
By using MS Excel, we have, at 672, the remainder is found as follows;
[tex]r(x) = \left(\dfrac{2015}{672} -2\right) \times (672) = 671[/tex]
At 672, [tex]\left(\dfrac{p(x) }{(x - a)} -q(x) \right)[/tex] ≈ 0.999, which when multiplied by 672, gives
the biggest remainder of 671, which is the biggest remainder.
- The biggest remainder obtained by dividing 2015 by either 1, 2, 3, 4,...,1000 is (C) 671
Learn more about the remainder theorem here:
https://brainly.com/question/3283462
