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

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

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New post 22 Feb 2018, 02:12
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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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Re: What is the units digit of 2^105+3^105+4^105+5^105?  [#permalink]

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New post 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
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Re: What is the units digit of 2^105+3^105+4^105+5^105?  [#permalink]

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New post 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
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Re: What is the units digit of 2^105+3^105+4^105+5^105?  [#permalink]

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New post 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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