The remainder when 19^ 200 +23^ 200 is divided by 49 , is .......… - Vidyadip

MCQ

The remainder when $19^{200}+23^{200}$ is divided by $49$ , is $.........$.

A
$28$
B
$27$
✓
$29$
D
$26$

✓

Answer

Correct option: C.

$29$

c
$(21+2)^{200}+(21-2)^{200}$

$\Rightarrow 2\left[{ }^{100} C _0 21^{200}+200 C _2 21^{198} \cdot 2^2+\ldots . .+{ }^{200} C _{198} 21^2\right.$

$\left.2^{198}+2^{200}\right]$

$\Rightarrow 2\left[49 I _1+2^{200}\right]=49 I _1+2^{201}$

Now, $2^{201}=(8)^{67}=(1+7)^{67}=49 I _2+{ }^{67} C _0{ }^{67} C _1 \cdot 7=$ $49 I _2+470=49 I _2+49 \times 9+29$

$\therefore$ Remainder is $29$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

A function $y = f (x) $ satisfies $(x + 1) . f ' (x) -2 (x^2 + x) f (x) = \frac{{{e^{{x^2}}}}}{{(x + 1)}}, \forall x >-1$ . If $f$ $(0)$ $=$ $5$ , then $f$ $(x)$ is

If $0 < x < 1$, then ${\cot ^{ - 1}}\left( {\frac{{2{x^2} - 1}}{{2x\sqrt {1 - {x^2}} }}} \right)$ is equal to

If $f'(x) = \frac{1}{x} + x$ and $f(1) = \frac{5}{2}$, then $f(x) = $

If $\sum_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to.

Position of the point $(1, 1)$ with respect to the circle ${x^2} + {y^2} - x + y - 1 = 0$ is

Find the mean deviation about the mean for the data.

Height in cms	Number of boys
$95-105$	$9$
$105-115$	$13$
$115-125$	$26$
$125-135$	$30$
$135-145$	$12$
$145-155$	$10$

The population $P = P ( t )$ at time ${ }^{\prime} t ^{\prime}$ of a certain species follows the differential equation $\frac{ dP }{ dt }=0.5 P -450 .$ If $P (0)=850,$ then the time at which population becomes zero is

Let $x_1,x_2,.........,x_{100}$ are $100$ observations such that $\sum {{x_i} = 0,\,\sum\limits_{1 \leqslant i \leqslant j \leqslant 100} {\left| {{x_i}{x_j}} \right|} } = 80000\,\& $ mean deviation from their mean is $5,$ then their standard deviation, is-

The equation of the circle whose diameters have the end points $(a, 0), (0, b)$ is given by

The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L: 9 x+5 y=$ 45 between the axes. If $m_1$ and $m_2$ are the slopes of the lines $l_1$ and $1_2$,then the point of intersection of the line $y =\left( m _1+ m _2\right) x$ with $L$ lies on