Question

If 13^143 + 143 is divided by 14, then the remainder is

Answer 1

SOLUTION

TO DETERMINE

The remainder when

$\sf{ {13}^{143} + 143}$

is divided by 14

EVALUATION

$\sf{ {13}^{143} + 143}$

$\sf{ = {(14 - 1)}^{143} + 143}$

$\sf{ = ({14 }^{143} - {}^{143}C_1 \: {14 }^{142} + {}^{143}C_1 \: {14 }^{141} - .... - 1)+ 143}$

$\sf{ = ({14 }^{143} - {}^{143}C_1 \: {14 }^{142} + {}^{143}C_1 \: {14 }^{141} - .... + 14 )+ 142}$

$\sf{ = 14({14 }^{142} - {}^{143}C_1 \: {14 }^{141} + {}^{143}C_1 \: {14 }^{141} - .... + 1 1)+ 2}$

Now

$\sf{ 14({14 }^{142} - {}^{143}C_1 \: {14 }^{141} + {}^{143}C_1 \: {14 }^{141} - .... + 1 1)}$

is divisible by 14

So the required Remainder = 2

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Answer 2

We have to find the remainder if 13¹⁴³ + 143 is divided by 14.

solution : 13¹⁴³ + 143 is divided by 14, right ?

it is clear that when we divide 143/14 we get remainder 3 so, remainder from 143 = 3

now we have to focus on 13¹⁴³.

13¹⁴³ = (14 - 1)¹⁴³

you should remember that,

if we divide a number (m × a ± b )ⁿ by m

remainder of (m × a ± b)ⁿ/m = remainder of bⁿ/m

so, 13¹⁴³ is divided by 14 = (14 - 1)¹⁴³ is divided by 14.

from above explanation,

reminder of (14 - 1)¹⁴³/14 = remainder of (-1)¹⁴³/14

we know, (-1)¹⁴³ = -1

so, remainder of (-1)¹⁴³/14 = -1

now the remainder of 13¹⁴³ + 143 = -1 + 3 = 2

Therefore 2 will be remainder if we divide 13¹⁴³ + 143 is divided by 14.

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