Question

plz provide solution​

Answer 1

Answer:

a.1

Step-by-step explanation:

the remainder obtained is 1

Answer 2

Question :---

• Find the Remainder when 16^(2016) is Divided by 9?

Solution :--

Lets Try to Solve the Problem in a simple way :---

16 can be written as = (4)²

So, Question becomes now = [ (4)² ]^2016

→ (4)^4032 [ as (a^b)^c = a^(bc) ]

or,

→ [(4)³]^(1344) [ as 3*1344 = 4032 ]

→ (64)^1344

So, we can write ,

=> 16^(2016) / 9 = (64)^1344 / 9

Now, we know that , when 64 is divided by 9 , we get remainder as 1 . ( 9*7 + 1)

So,

→ (64)^1344 / 9

→ (1)^1344/ 9

→ 1 Remainder .. (a) (Ans)..

______________________________

Lets Try To solve it with Euler's Theoram Now :---

Euler’s theorem states that if p and n are coprime positive integers, than,

P φ (n) = 1 (mod n), where Φn = n*(1-1/a)*(1-1/b)

→ 16^(2016) /9

16 and 9 are Relatively Prime number, So,

→ φ (9) = 9 * ( 1- 1/3)

→ φ (9) = 9 * 2/3

φ (9) = 6

Hence ,

→ 16^6 = 1 (mod 9) .

Now, Question Can be written as :---

→ (16)^2016 = 1 (mod 9)

Or,

→ (16^6)^336 = 1 (mod 9)

Hence, The remainder is 1 ...