The remainder on dividing 599 by 11 is \_\_\_\_\_\_\_\_.
Answer: 9
Solutions · 1
Official solution
\[5^{99}=5^4\cdot5^{95}=625\left[5^5\right]^{19}=625[3125]^{19}=625[3124+1]^{19}\]
Since 3124=11×284, [3124+1]19=11k+1 for some integer k.
\[5^{99}=625(11k+1)=625\times11k+625=11k_1+616+9=11k_2+9\]
Remainder = 9