Question

What is the remainder when ${82}^{361}+{83}^{361}+{84}^{361}+{85}^{361}+{86}^{361}$ is divided by 7 ?

Answer 1

${82}^{361}+{83}^{361}+{84}^{361}+{85}^{361}+{86}^{361}=\left[\right(84-2{)}^{361}+(84-1{)}^{361}+{84}^{361}+(84+1{)}^{361}+(84+2{)}^{361}]$
Since 84 is a multiple of 7 then the remainder will be when $(-2{)}^{361}+(-1{)}^{361}+1^{361}+2^{361}$ is divided by 7
is $(-2{)}^{361}+(-1{)}^{361}+1^{361}+2^{361}=0$ So the remainder is zero