Can someone answer this
2 answers:
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The answer is 217 with a remainder of 20.
explanation is in the picture.
Observe that
Now,
so that
To decide which is the correct value, we need to examine the sign of . It evaluates to 0 if
We have
Also,
and increases as
increases, which means
Therefore for all ,
For example, when , we get
Then the target expression has a negative sign at the given value of :
Alternatively, we can try simplifying by denesting the radical. Let
be non-zero integers (
) such that
Note that the left side must be positive.
Taking squares on both sides gives
Let and
. Then
Only the first case leads to integer coefficients. Since , one of
or
must be negative. We have
Now if , then
, and
However, , so
is negative, so we don't want this.
Instead, if , then
, and thus
Then our target expression evaluates to