What is the remainder when (2^469 + 3^268) is divided by 22?
(1) 1
(2) 11
(3) 19
(4) 0
(5) 17
Answers
when 20469+30268 ÷22
Answer:
Answer is 11.
Step-by-step explanation:
Remainder when 2^469 + 3^268 is divided by 22
We need to solve this by finding the remainder individually.
2^469 / 22 = 2^468/11 = 2^(5*93 + 3)/11 = (2^(5 * 93) * 2^3) / 11
= (32^93)(8) = (8*(33 – 1)^93)/11
In (33 – 1)^93, every term will have 33 except the last term which is -1.
That means It is in the form of (33x – 1)
8*(33X – 1)/11 = (33*X*8 – 8)/11
The above expression always gives remainder 3 for X > 1.
Hence remainder when divided with 11 is 3.
If we divide with 22, the final reminder of first expression will be 6.---------------E1.
3^268/22 = 3^(265 + 3)/22 = (3^(5*53) * 3^3)/22
= (243^53)*27/22 = 27*(242 + 1)^53 / 22
In (242 + 1)^ 53, every term has 242 except last term which is 1.
So we can rewrite that expression as 27(242X + 1)/22
Above expression always gives the remainder 27/22 (242 is divided by 22)
Hence remainder of second expression is 5. ------------------------------E2
From E1 and E2, we can say that final expression remainder is 6+5 = 11.