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.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

well the answer is 3+2 = 5 .

well the answer is 3+2 = 5 . check if you have done any calculation mistake

the answer is 3 + 2 = 5 .

the answer is 3 + 2 = 5 . check for calculation mistakes

i have a better solution i

i have a better solution i guess. needs some insight though. 17 = 5 + 12 and 12 + 13 = 25 with which i am dividing thus 17^19 + 13^19 = (5+12)^19 + 13^19 binomially expanding the first term leads to : (terms divisible by 5^2 = 25) + 5*19C18* 12^18 + 12^19 thus 17^19 + 13^19 = 25k + 5*19*12^18 + (12^19 + 13^19 ) now (12^19 + 13^19 ) is divisible by (12+13) = 25. thus the remainder term will be obtained from 5*19*12^18 taking 5 common out, lets find the remainder when 19*12^18 is divided by 5. the power residues of 12 mod 5 are cyclic with periodicity 4 from fermat's theorem. thus 12^18 has the same remander as 12^2 = 4. the remainder when 19 * 4 is divided by 5 = 1. thus: 19*12^18 = 5m + 1 thus 5*19*12^18 = 25 m + 5. HENCE THE REMAINDER IS 5.

Hey Rajorshi again you have

Hey Rajorshi again you have shown a better method and new concept Thanks and hopefully we ‘ll keep learning from you. It took some time in understanding the solution but it wroth the time