If λ is the remainder when 22021 is divided by 17, then the value of λ must be equal to

Question

If λ is the remainder when 22021 is divided by 17, then the value of λ must be equal to (A) 3 (B) 7 (C) 13 (D) 15

Tardigrade Admin · Accepted Answer

22021=2.22020=2.(24)505=2(16)505=2(17 - 1)505 =2 1 7505 - 505 C1 1 7504 + 505 C2 1 7503 . . . . . . . . . . . . + 505 C504 1 71 - 505 C505 1 =2 17 k - 1 =17⋅ 2k-2=17⋅ 2k-17+15 =17(2 k - 1)+15

Q. If $λ$ is the remainder when $2^{2021}$ is divided by $17,$ then the value of $λ$ must be equal to

$3$

$7$

$13$

$15$

$2^{2021} = 2. 2^{2020} = 2. (2^{4})^{505} = 2 (16)^{505} = 2 (17 - 1)^{505}$
$= 2 {1 7^{505} -^{505} C_{1} 1 7^{504} +^{505} C_{2} 1 7^{503} ............ +^{505} C_{504} 1 7^{1} -^{505} C_{505} 1}$
$= 2 {17 k - 1}$
$= 17 \cdot 2 k - 2 = 17 \cdot 2 k - 17 + 15$
$= 17 (2 k - 1) + 15$

Q. If λ is the remainder when 22021 is divided by 17, then the value of λ must be equal to

3

7

13

15

22021=2.22020=2.(24)505=2(16)505=2(17−1)505 =2{17505−505C1​17504+505C2​17503............+505C504​171−505C505​1} =2{17k−1} =17⋅2k−2=17⋅2k−17+15 =17(2k−1)+15

Q. If $λ$ is the remainder when $2^{2021}$ is divided by $17,$ then the value of $λ$ must be equal to

$3$

$7$

$13$

$15$

$2^{2021} = 2. 2^{2020} = 2. (2^{4})^{505} = 2 (16)^{505} = 2 (17 - 1)^{505}$
$= 2 {1 7^{505} -^{505} C_{1} 1 7^{504} +^{505} C_{2} 1 7^{503} ............ +^{505} C_{504} 1 7^{1} -^{505} C_{505} 1}$
$= 2 {17 k - 1}$
$= 17 \cdot 2 k - 2 = 17 \cdot 2 k - 17 + 15$
$= 17 (2 k - 1) + 15$