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What is the unit digit of 3^5+4^5+5^5+6^5?

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What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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New post 04 Sep 2017, 02:06
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What is the unit digit of \(3^5+4^5+5^5+6^5\)?

A. 5
B. 6
C. 7
D. 8
E. 9

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Re: What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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New post 04 Sep 2017, 02:22
MathRevolution wrote:
What is the unit digit of \(3^5+4^5+5^5+6^5\)?

A. 5
B. 6
C. 7
D. 8
E. 9


\(3^5 = 3^4 * 3 = 81 * 3 = (...3)\)

\(4^5 = 4^4 * 4 = (4^2)^2 * 4 = (...6)^2 * 4 = (...6) * 4 = (...4)\)

\(5^5 = (5^2)^2 * 5 = (...5)^2 * 5 = (...5) * 5 = (...5)\)

\(6^5 = (...6)\)

Hence the unit digit will be

\((...3) + (...4) + (...5) + (...6) = (...8)\)

Answer D.
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What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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New post 04 Sep 2017, 08:02
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MathRevolution wrote:
What is the unit digit of \(3^5+4^5+5^5+6^5\)?

A. 5
B. 6
C. 7
D. 8
E. 9

\(3^5+4^5+5^5+6^5\)

Question can be solved by using Cyclicity of power properties. We need to find Cyclicity of power of each number. Then divide the exponent by the cyclicity of power and the remainder becomes the new exponent.

Cyclicity of power of \(3\) is \(4 : (3,9,7,1). \frac{5}{4}\) will give remainder \(1\). Therefore \(3^5 = 3^1\). Hence Unit digit of \(3^5 = 3\)

Cyclicity of power of \(4\) is \(2 : (4,6). \frac{5}{2}\) will give remainder \(1\). Therefore \(4^5 = 4^1\). Hence Unit digit of \(4^5 = 4\)

Cyclicity of power of \(5\) is \(1 : (5). 5\) raised to any exponent will have unit digit as \(5\). Hence Unit digit of \(5^5 = 5\)

Cyclicity of power of \(6\) is \(1\): (\(6\)). \(6\) raised to any exponent will have unit digit as \(6\). Hence Unit digit of \(6^5 = 6\)

\(3 + 4 + 5 + 6 = 18\)

Therefore Unit digit is \(8\)

Answer (D)...


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Re: What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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New post 04 Sep 2017, 08:24
Numbers 2,3,7,8 have a cyclicity of 4.
Numbers 4 and 9 have a cyclicity of 2.
Numbers 1,5,6 have a cyclicity of 1.

3^5 will have the same units digit of 3^1(3)
Similarly, 4^5 has the same units digit of 4^1(4)
Since 5 and 6 both have a cyclicity of 1, each power of 5 and 6 have the same units digit.

From this information \(3^5+4^5+5^5+6^5\) will have units digit 3,4,5,6 which when added give a units digit of 8(Option D)
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Re: What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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New post 06 Sep 2017, 01:33
=> 3^1 has a unit digit 3.
3^2 has a unit digit 9.
3^3 has a unit digit 7.
3^4 has a unit digit 1.
3^5 has a unit digit 3.

4^1 has a unit digit 4.
4^2 has a unit digit 6.
4^3 has a unit digit 4.
4^4 has a unit digit 6.
4^5 has a unit digit 4.

5^1 has a unit digit 5.
5^2 has a unit digit 5.
5^n has a unit digit 5 for all positive integers n.

6^1 has a unit digit 6.
6^2 has a unit digit 6.
6^n has a unit digit 6 for positive integers n.

Thus 3 + 4 + 5 + 6 = 18 has the unit digit 8.

Ans: D
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Re: What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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New post 06 Sep 2017, 02:31
MathRevolution wrote:
What is the unit digit of \(3^5+4^5+5^5+6^5\)?

A. 5
B. 6
C. 7
D. 8
E. 9


the cyclicity of 3 is 4. So 3^5 will have a unit's digit of 3

the cyclicity of 4 is 2. So 4^5 will have a unit's digit of 4

the cyclicity of 5 is 1. So 5^5 will have a unit's digit of 5

the cyclicity of 6 is 1. So 3^5 will have a unit's digit of 6

Adding the unit's digit of all we get 3+4+5+6 = 18

So, unit's digit of \(3^5+4^5+5^5+6^5\) = 8.

Answer D
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