Binomial theorem for positive integral Index

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$2^{60}$ when divided by $7$ leaves the remainder

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Solution

We have,
$\begin{array}{l} 2^{60} = (1 + 7)^{20} \\ \Rightarrow 2^{60} =^{20} C_{0} +^{20} C_{1} \times 7 +^{20} C_{2} \times 7^{2} + \dots +^{20} C_{20} \times 7^{20} \\ \Rightarrow 2^{60} = 7 (^{20} C_{1} +^{20} C_{2} \times 7 + \dots +^{20} C_{20} \times 7^{19}) + 1 \end{array}$
Hence, $2^{60}$ when divided by $7$ leaves the remainder $1$ .

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Binomial theorem for positive integral Index

Remember concepts with our Masterclasses.

260 when divided by 7 leaves the remainder

We have, 260=(1+7)20⇒ 260=20C0+20C1×7+20C2×72+…+20C20×720⇒ 260=7 20C1+20C2×7+…+20C20×719+1Hence, 260 when divided by 7 leaves the remainder 1.

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$2^{60}$ when divided by $7$ leaves the remainder