Go To Your Study Dashboard

Get Answers to all your Questions

header-bg

When $32^{\left(32^{32)}\right.}$ is divided by $7,$ the remainder is ______ .
Option: 1
$2$

Option: 2
$3$

Option: 3
$4$

Option: 4
$9$

Answers (1)

We have $32^{32}=\left(2^5\right)^{32}=2^{160}$

$=(3-1)^{160}={ }^{160} C_0 3^{160}-{ }^{160} C_1 3^{159}+\ldots-{ }^{160} C_{159} \cdot 3+1$

$=3 m+1,$ where $m$ is some positive integer.
Now, $32^{\left(32^{32)}\right.}=\left(2^5\right)^{3 m+1}=\left(2^3\right)^{5 m+1} \cdot 2^2=(7+1)^{5 m+1} \cdot 4$

But $(7+1)^{5 m+1}=7 n+1$ for some positive integer $n$ . Thus, $N=32^{\left(32^{32)}\right.}=28 n+4 .$

Thus shows that when $N$ is divided by $7,$ the remainder is 4 $4.$

Posted by

Kshitij

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

50 mL of 0.2 M ammonia solution is treated with 25 mL of 0.2 M HCl. If pK_b of ammonia solution is 4.75, the pH of the mixture will be :
Option: 1 3.75
Option: 2 4.75

Trending Articles/News

anam-khan

BTech aspirant from north-east or UT? Apply for CSAB NEUT counselling 2025 today

anam-khan

JoSAA round 2 cut-off 2025 out; BTech closing ranks for top engineering courses at IITs

anam-khan

Goa Engineering Admission 2025: 1,415 eligible for BE seats; merit list on June 27

Latest Question