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# What is the units digit of 2^105+3^105+4^105+5^105?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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What is the units digit of 2^105+3^105+4^105+5^105?  [#permalink]

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22 Feb 2018, 02:12
00:00

Difficulty:

15% (low)

Question Stats:

79% (01:31) correct 21% (01:50) wrong based on 77 sessions

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[GMAT math practice question]

What is the units digit of $$2^{105}+3^{105}+4^{105}+5^{105}?$$

$$A. 3$$
$$B. 4$$
$$C. 5$$
$$D. 6$$
$$E. 7$$

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Current Student Joined: 07 Jan 2016 Posts: 1086 Location: India GMAT 1: 710 Q49 V36 Re: What is the units digit of 2^105+3^105+4^105+5^105? [#permalink] ### Show Tags 22 Feb 2018, 03:52 1 MathRevolution wrote: [GMAT math practice question] What is the units digit of $$2^{105}+3^{105}+4^{105}+5^{105}?$$ $$A. 3$$ $$B. 4$$ $$C. 5$$ $$D. 6$$ $$E. 7$$ last digits of squares of 2 has a cycle of 4 i.e 2 , 4 , 8, 6 , 2 , 4 , 8 , 6 3 has a cycle of 4 i.e 3,9, 7,1 , 3 , 9 , 7, 1 4 has a cycle of 2 i.e 4,6,4,6,4 5 has a cycle of 5 i.e 1 now 105 in the form 4k+1 or 2k+1 so 2^105 will have same digit as 2^1 similar for 3,4,5 2+3+4+5 = 14 ends with a 4 (B) imo Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8017 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the units digit of 2^105+3^105+4^105+5^105? [#permalink] ### Show Tags 25 Feb 2018, 18:42 => The units digit of any integer raised to the exponent $$5$$ is the same as the units digit of the integer: $$0^1$$ and $$0^5$$ have remainder $$0$$ $$1^1$$ and $$1^5$$ have remainder $$1$$ $$2^1$$ and $$2^5$$ have remainder $$2$$ $$9^1$$ and $$9^5$$ have remainder $$9$$ when divided by $$10$$. Therefore, the units digit of $$2^{105}+3^{105}+4^{105}+5^{105}$$ is same as the units digit of $$2 + 3 + 4 + 5 = 14.$$ It is $$4$$ Therefore, B is the answer. Answer: B _________________ 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$79 for 1 month Online Course"
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Re: What is the units digit of 2^105+3^105+4^105+5^105?  [#permalink]

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14 Oct 2019, 11:21
MathRevolution wrote:
[GMAT math practice question]

What is the units digit of $$2^{105}+3^{105}+4^{105}+5^{105}?$$

$$A. 3$$
$$B. 4$$
$$C. 5$$
$$D. 6$$
$$E. 7$$

$$2^{105}+3^{105}+4^{105}+5^{105}?$$

= $$2^{26*4+1}+3^{105}+4^{26*4+1}+5^{105}?$$

= $$2 + 7 + 5$$

= $$4$$ , Answer must be (B) 4
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Re: What is the units digit of 2^105+3^105+4^105+5^105?   [#permalink] 14 Oct 2019, 11:21
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