Question

Maria divides 2015 by 1. Then she divides 2015 by 2 and then in order by 3, 4 etc. up to and including 1000. For each division she writes down the remainder. What is the biggest remainder she has noted down?
(A) 15 (B) 215
(C) 671 (D) 1007 (E) another value

Answer 1

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;

$\dfrac{r(x) }{(x - a)} = \dfrac{p(x) }{(x - a)} -\dfrac{(x - a) \times q(x) }{(x - a)} = \mathbf{ \dfrac{p(x) }{(x - a)} -q(x)}$

Therefore;

$r(x) = \mathbf{\left(\dfrac{p(x) }{(x - a)} -q(x) \right) \times (x - a)}$

The remainder is largest when both  $\left(\dfrac{p(x) }{(x - a)} -q(x) \right)$ and (x - a) are large

$\left(\dfrac{p(x) }{(x - a)} -q(x) \right)$ 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;

$r(x) = \left(\dfrac{2015}{672} -2\right) \times (672) = 671$

At 672,  $\left(\dfrac{p(x) }{(x - a)} -q(x) \right)$ ≈ 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:

brainly.com/question/3283462