## will clear many concepts...

.What is the remainder when 17^19 + 13^19 is divided by 25?

## One approach

**Reminder of ( a + b )/m = Reminder of a/m + Reminder of b/m**

Reminder of 17^19 + 13^19 = Reminder of 17^19 + Reminder of 13^19
Reminder of 17^19 divided by 25
= Reminder of { 17 x (289)^9 } divided by 25
= Reminder of 17 x Reminder of { (14)^9 } divided by 25
= Reminder of (17 x 14) x Reminder of { (196)^4 } divided by 25
= Reminder of 17 x 14 x [ Reminder of { (21)^4 } divided by 25 ]
= Reminder of 17 x 14 x [ Reminder of { (441)^2 } divided by 25 ]
= Reminder of 17 x 14 x [ Reminder of { (16)^2 } divided by 25 ]
= Reminder of 17 x 14 x 6 divided by 25
= Reminder of 17 x 84 divided by 25
= Reminder of 17 x 9 divided by 25
= Reminder of 153 divided by 25
= 3
Similarly, Reminder of 13^19 divided by 25
= Reminder of { 13 x (169)^9 } divided by 25
= 13 x Reminder of { (19)^9 } divided by 25
= (13 x 19 ) x Reminder of { (19)^8 } divided by 25
= (13 x 19 ) x Reminder of { (381)^4 } divided by 25
= (13 x 19 ) x Reminder of { (6)^4 } divided by 25
= (13 x 19 ) x Reminder of { (36)^2 } divided by 25
= (13 x 19 ) x Reminder of { (11)^2 } divided by 25
= (13 x 19 ) x Reminder of (121) divided by 25
= Reminder of 13 x 19 x 21 divided by 25
= Reminder of 247 x 21 divided by 25
= Reminder of 22 x 21 divided by 25
= 12
Total reminder = 3 + 12 = 15