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

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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04 Sep 2017, 02:06
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5% (low)

Question Stats:

95% (01:16) correct 5% (01:13) wrong based on 60 sessions

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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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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Senior CR Moderator Status: Long way to go! Joined: 10 Oct 2016 Posts: 1354 Location: Viet Nam Re: What is the unit digit of 3^5+4^5+5^5+6^5? [#permalink] ### Show Tags 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. _________________ Director Joined: 04 Dec 2015 Posts: 750 Location: India Concentration: Technology, Strategy Schools: ISB '19, IIMA , IIMB, XLRI WE: Information Technology (Consulting) What is the unit digit of 3^5+4^5+5^5+6^5? [#permalink] ### Show Tags 04 Sep 2017, 08:02 1 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)... _________________ Please Press "+1 Kudos" to appreciate. Senior PS Moderator Joined: 26 Feb 2016 Posts: 3386 Location: India GPA: 3.12 Re: What is the unit digit of 3^5+4^5+5^5+6^5? [#permalink] ### Show Tags 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) _________________ You've got what it takes, but it will take everything you've got Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7224 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the unit digit of 3^5+4^5+5^5+6^5? [#permalink] ### Show Tags 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
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Re: What is the unit digit of 3^5+4^5+5^5+6^5?  [#permalink]

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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.

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Re: What is the unit digit of 3^5+4^5+5^5+6^5?   [#permalink] 06 Sep 2017, 02:31
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