Question

what is the remainder when 7^1287 is divided by 172 ?​

Answer 1

Answer:

And answer is 111 hope ita right

Answer 2

SOLUTION

TO DETERMINE

The remainder when

$\sf{ {7}^{1287} }$

is divided by 172

EVALUATION

Here the given number is

$\sf{ {7}^{1287} }$

Now

$\sf{ {7}^{1287} }$

$\sf{ = {( {7}^{3} )}^{429} }$

$\sf{ = {( 343 )}^{429} }$

$\sf{ = {( 344 - 1 )}^{429} }$

$\sf{ = {( 344)}^{429} - {}^{429}C_1 \: {( 344)}^{428} + {}^{429}C_2 \: {( 344)}^{427} - ... + {}^{429}C_{428} \: {( 344)}^{1} - {}^{429}C_{429} \: }$

$\sf{ = {( 344)}^{429} - {}^{429}C_1 \: {( 344)}^{428} + {}^{429}C_2 \: {( 344)}^{427} - ... + {}^{429}C_{428} \: {( 344)}^{1} - 1 }$

Now each of

$\sf{ {( 344)}^{429} - {}^{429}C_1 \: {( 344)}^{428} + {}^{429}C_2 \: {( 344)}^{427} - ... + {}^{429}C_{428} \: {( 344)}^{1} \: }$

is divisible by 172

So the required Remainder = 172 - 1 = 171

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