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Re: What is the remainder when 13^17 + 17^13 is divided by 10?
[#permalink]
17 Aug 2018, 16:05
6
1
Expert Reply
Explanation
The remainder when dividing an integer by 10 always equals the units digit. You can also ignore all but the units digits, so the question can be rephrased as: What is the units digit of \(3^{17} + 7^{13}\)?
The pattern for the units digits of 3 is [3, 9, 7, 1]. Every fourth term is the same. The 17th power is 1 past the end of the repeat: 17 – 16 = 1. Thus, \(3^{17}\) must end in 3.
The pattern for the units digits of 7 is [7, 9, 3, 1]. Every fourth term is the same. The 13th power is 1 past the end of the repeat: 13 – 12 = 1. Thus, \(7^{13}\) must end in 7. The sum of these units digits is 3 + 7 = 10. Thus, the units digit is 0.
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Re: What is the remainder when 13^17 + 17^13 is divided by 10?
[#permalink]
16 Aug 2018, 00:39
4
Remainder Property \(a^n\) + \(b^n\) is divisible by a+b if n is odd. Here n=17 and 13 (odd). Now, a+b = 17+13 =30. Which is completely divisible by 10. Hence remainder is zero.
Similarly, if we calculate the pattern for 13, we'll observe that it repeats in a similar way 13 does, after 4 multiplications. So remainders will be (3+7)%10=0
Re: What is the remainder when 13^17 + 17^13 is divided by 10?
[#permalink]
16 Sep 2022, 08:45
1
We need to find what is the remainder when \(13^{17} + 17^{13}\) 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 \(13^{17} + 17^{13}\) by 10 = Remainder of \(13^{17}\) by 10 + Remainder of \(17^{13}\) by 10
Unit's digit of \(13^{17}\)
= Unit's digit of \(3^{17}\)
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 \(3^1\) = 3 Unit's digit of \(3^2\) = 9 Unit's digit of \(3^3\) = 7 Unit's digit of \(3^4\) = 1 Unit's digit of \(3^5\) = 3
So, unit's digit of power of 3 repeats after every \(4^{th}\) number. => We need to divided 17 by 4 and check what is the remainder => 17 divided by 4 gives 1 remainder
=> \(3^{17}\) will have the same unit's digit as \(3^1\) = 3 => Unit's digits of \(13^{17}\) = 3
Unit's digit of \(17^{13}\)
= Unit's digit of \(7^{13}\)
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 13 by 4 and check what is the remainder => 13 divided by 4 gives 1 remainder
=> \(7^{13}\) will have the same unit's digit as \(7^1\) = 7 => Unit's digits of \(17^{13}\) = 7
=> Unit's digits of \(13^{17}\) + Unit's digits of \(17^{13}\) = 3 + 7 = 10
But remainder of \(13^{17} + 17^{13}\) 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
Re: What is the remainder when 13^17 + 17^13 is divided by 10?
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10 Dec 2023, 20:13
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