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Re: What is the remainder when 13^17 + 17^13 is divided by 10?
[#permalink]
16 Sep 2022, 08:41
We need to find what is the remainder when 1317+1713 is divided by 10
Theory: Remainder of sum of two numbers = Sum of their individual remainders Remainder of any number by 10 = Unit's digit of that number
=> Remainder of 1317+1713 by 10 = Remainder of 1317 by 10 + Remainder of 1713 by 10
Unit's digit of 1317
= Unit's digit of 317
We can do this by finding the pattern / cycle of unit's digit of power of 3 and then generalizing it.
Unit's digit of 31 = 3 Unit's digit of 32 = 9 Unit's digit of 33 = 7 Unit's digit of 34 = 1 Unit's digit of 35 = 3
So, unit's digit of power of 3 repeats after every 4th number. => We need to divided 17 by 4 and check what is the remainder => 17 divided by 4 gives 1 remainder
=> 317 will have the same unit's digit as 31 = 3 => Unit's digits of 1317 = 3
Unit's digit of 1713
= Unit's digit of 713
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 71 = 7 Unit's digit of 72 = 9 Unit's digit of 73 = 3 Unit's digit of 74 = 1 Unit's digit of 75 = 7
So, unit's digit of power of 7 repeats after every 4th number. => We need to divided 13 by 4 and check what is the remainder => 13 divided by 4 gives 1 remainder
=> 713 will have the same unit's digit as 71 = 7 => Unit's digits of 1713 = 7
=> Unit's digits of 1317 + Unit's digits of 1713 = 3 + 7 = 10
But remainder of 1317+1713 by 10 cannot be more than or equal to 10 => Remainder = Remainder of 10 by 10 = 0
So, Answer will be 0 Hope it helps!
Watch the following video to learn the Basics of Remainders
gmatclubot
Re: What is the remainder when 13^17 + 17^13 is divided by 10? [#permalink]