1. Binomial Theorem Method
(6^83+8^83)%49= ((7-1)^83 + (7+1)^83)%49
Expand (a-b)^n and (a+b)^n
You will notice that all the terms would be divisible by 49 except last 2,
since all the terms would have power of 7 more than 2 except the last 2 which would be 7^1 and 7^0 (not divisible by 49).
So we r left with 83C82 * 7 - 1 + 83C82 * 7 + 1(last two terms of both expansion)
The question now boils down to (2*83*7) % 49
Divide by 7 ==> Question becomes (2*83)%7 viz. 166%7=5
Since we divided by 7, we have to multiply the intermediate remainder by 7.
Therefore, The Final Remainder = 5 * 7 = 35
(6^83+8^83)%49= ((7-1)^83 + (7+1)^83)%49
Expand (a-b)^n and (a+b)^n
You will notice that all the terms would be divisible by 49 except last 2,
since all the terms would have power of 7 more than 2 except the last 2 which would be 7^1 and 7^0 (not divisible by 49).
So we r left with 83C82 * 7 - 1 + 83C82 * 7 + 1(last two terms of both expansion)
The question now boils down to (2*83*7) % 49
Divide by 7 ==> Question becomes (2*83)%7 viz. 166%7=5
Since we divided by 7, we have to multiply the intermediate remainder by 7.
Therefore, The Final Remainder = 5 * 7 = 35
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