2 to the power 405 divided by 7 then what is the quotient and remainder

first u have to find 2 to the power 405=? find the value

If u need to find the remainder the remainder , theres no need to evaluate 2^405 , this can be solved in the following ways.

2405 = (8)135=(7+1)135

Binomially expand the above expression as

= ¹³⁵C₀7¹³⁵+¹³⁵C₁7¹³⁴+¹³⁵C₂7¹³³+...+¹³⁵C₁₃₄7¹+¹³⁵C₁₃₅7⁰

= In the above expansion, all the terms are divisible by 7 (They consist of 7 as a factor)

= Only ¹³⁵C₁₃₅ 7⁰ = 1 is non divisible as such 1 is the remainder

If you wanna find the quotient  try logarithm

2⁴⁰⁵= antilog(405 log2) , it will give u a fair approx. values

Quotient = antilog(405*log2 - log7 )

The remainder is 1....................

2^405 mod 7

(2^5)^81 ?mod7

(4)^81 mod7

(4^3)^27 mod7

(1)^27 mod 7

1 mod7

Therefore remainder is 1............