The remainder on dividing $5^{99}$ by 11 is ____________.
Answer (integer)
9
Solution
$5^{99}=5^{4} .5^{95}$
<br/><br/>$=625\left[5^{5}\right]^{19}$
<br/><br/>$=625[3125]^{19}$
<br/><br/>$=625[3124+1]^{19}$
<br/><br/>$=625[11 \mathrm{k} \times 19+1]$
<br/><br/>$=625 \times 11 \mathrm{k} \times 19+625$
<br/><br/>$=11 \mathrm{k}_{1}+616+9$
<br/><br/>$=11\left(\mathrm{k}_{2}\right)+9$
<br/><br/>Remainder $=9$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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