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Question
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Solution
\[3^{3n} - 26n - 1 = {27}^n - 26n - 1 . . . \left( 1 \right)\]
\[\text{ Now, we have: } \]
\[ {27}^n = (1 + 26 )^n \]
\[\text{ On expanding, we get } \]
\[(1 + 26 )^n = ^{n}{}{C}_0 \times {26}^0 +^{n}{}{C}_1 \times {26}^1 + ^{n}{}{C}_2 \times {26}^2 + ^{n}{}{C}_3 \times {26}^3 +^{n}{}{C}_4 \times {26}^4 + . . . ^{n}{}{C}_n \times {26}^n \]
\[ \Rightarrow {27}^n = 1 + 26n + {26}^2 [^{n}{}{C}_2 + ^{n}{}{C}_3 \times {26}^1 + ^{n}{}{C}_4 \times {26}^2 + . . . ^{n}{}{C}_n \times {26}^{n - 2} ]\]
\[ \Rightarrow {27}^n - 26n - 1 = 676 \times \text{ an integer } \]
\[ {27}^n - 26n - 1 \text{ is divisible by } 676\]
\[\text{ Or, }\]
\[ 3^{3n} - 26n - 1 \text{ is divisible by } 676 \left( \text{ From } (1) \right)\]
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