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  4. The remainder on dividing 599 by 11 is

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Q. The remainder on dividing $5^{99}$ by 11 is

JEE MainJEE Main 2023Binomial Theorem

Solution:

$ 5^{9 9}=5^4 .5^{95} $
$ =625\left[5^5\right]^{19} $
$ =625[3125]^{19} $
$ =625[3124+1]^{19} $
$ =625[11 k \times 19+1] $
$ =625 \times 11 k \times 19+625$
$ =11 k _1+616+9$
$ =11\left( k _2\right)+9 $
Remainder $=9$