Question

what is the remainder if the number 7^2015 is divided by 25

Answer 1

Hope it helps!!!
(WARNING!!) The above solution is for high school students.
And the next solution is for advanced high school students or for students planning to take part in
RMO, INMO, IMO.
My friend, there is an easy and beautiful function which will help you to crack any problem on modulo arithmetic…-the euler totient function-you can refer it in the net to know more …,it is noted by phi..

so here, phi of 25 is 20 ,and 7^2015=7^2000 .7^15=(7^20)^100 .7^15,by using the function we can say all we need to find is 7^15 % 25. so (7^2)^7 . 7 % 25

=49^7 . 7 % 25

=-1^7 . 7 % 25 (as 49 %25 =-1,also by using ab % c =a%c * b%c)

=-7 % 25=25 -7 =18…

maths is the singularity of the black hole beauty…..enjoy maths!!

Answer 2

iitian have already solved it by two methods I have third one so I am presenting it
by using •binomial expansion•
7^2015 = 7*7^2014= 7(49)^1007
=7(50-1)^1007
on expansion we get
=7(1007c0 50^1007 (-1)^0 + 1007c1 50^1006 (-1)^1
+......... + 1007c1006 50^1 (-1)^1006 +
1007c1007 50^0 (-1)^1007)
thus we saw that there is 50 in each term except the last one thus all terms are divisible by 25 excepting last one
so we saw that last term came out as -7
for remainder
-7-(-25) = -7+25 = 18
thus remainder is 18