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We know to find what is the remainder when \(43717^{(43628232)}\) is divided by 5
Theory: Remainder of a number by 5 is same as the reminder of the unit's digit of the number by 5
(Watch this Video to Learn How to find Remainders of Numbers by 5)
Using Above theory , Let's find the unit's digit of \(43717^{(43628232)}\) first.
Units' digit of \(43717^{(43628232)}\) = Same as units' digit of \(7^{(43628232)}\)
We can do this by finding the pattern / cycle of unit's digit of power of 7 and then generalizing it.
Unit's digit of \(7^1\) = 7
Unit's digit of \(7^2\) = 9
Unit's digit of \(7^3\) = 3
Unit's digit of \(7^4\) = 1
Unit's digit of \(7^5\) = 7
So, unit's digit of power of 7 repeats after every \(4^{th}\) number.
=> We need to divided 43628232 by 4 and check what is the remainder
=> 43628232 divided by 4 gives 0 remainder
=> \(7^{43628232}\) will have the same unit's digit as \(7^{Cycle}\) = \(7^{4}\) = 1
=> Remainder of \(7^{ 43628232 }\) by 5 = Remainder of 1 by 5 = 1
So, Answer will be A
Hope it helps!
MASTER How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem
Theory: Remainder of a number by 5 is same as the reminder of the unit's digit of the number by 5
(Watch this Video to Learn How to find Remainders of Numbers by 5)
Using Above theory , Let's find the unit's digit of \(43717^{(43628232)}\) first.
Units' digit of \(43717^{(43628232)}\) = Same as units' digit of \(7^{(43628232)}\)
We can do this by finding the pattern / cycle of unit's digit of power of 7 and then generalizing it.
Unit's digit of \(7^1\) = 7
Unit's digit of \(7^2\) = 9
Unit's digit of \(7^3\) = 3
Unit's digit of \(7^4\) = 1
Unit's digit of \(7^5\) = 7
So, unit's digit of power of 7 repeats after every \(4^{th}\) number.
=> We need to divided 43628232 by 4 and check what is the remainder
=> 43628232 divided by 4 gives 0 remainder
=> \(7^{43628232}\) will have the same unit's digit as \(7^{Cycle}\) = \(7^{4}\) = 1
=> Remainder of \(7^{ 43628232 }\) by 5 = Remainder of 1 by 5 = 1
So, Answer will be A
Hope it helps!
MASTER How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem
