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M12-29

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Bunuel
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M12-29 [#permalink] New post   16 Sep 2014, 00:47
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What is the unit's digit of \(7^{75} + 6\)?

A. 1
B. 3
C. 5
D. 7
E. 9
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E
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Bunuel
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Re M12-29 [#permalink] New post   16 Sep 2014, 00:47
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Official Solution:

What is the unit's digit of \(7^{75} + 6\)?

A. 1
B. 3
C. 5
D. 7
E. 9


The unit's digit of 7 raised to a positive integer power cycles in a pattern of four: 7-9-3-1. Given that \(75 = 4 *18 + 3\), the unit's digit of \(7^{75}\) aligns with the unit's digit of \(7^3\), which is 3.

Consequently, the unit's digit of \(7^{75} + 6\) is: 3 + 6 = 9.


Answer: E
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erikvm
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Re: M12-29 [#permalink] New post   05 Mar 2015, 08:44
Bunuel wrote:
Official Solution:




The unit's digit of 7 in a positive integer power repeats in blocks of 4: 7-9-3-1. Since \(75=4*18+3\) then the unit's digit of \(7^{75}\) is the same as the unit's digit of \(7^3\), which is 3.

Therefore, the unit's digit of \(7^{75} + 6\) will be: 3 plus 6 = 9.


Answer: E


I don't understand that. I assumed that 75 is a multiple of 5, and 7^5 ends in 7, so I 7 + 6 = unit digit of 3. Where does that 4*18 + 3 come from? I mean.. I could simply say 75 = 9*8 + 3? Is that how you decide what unit digit it is?
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Re: M12-29 [#permalink] New post   05 Mar 2015, 08:51
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erikvm wrote:
Bunuel wrote:
Official Solution:




The unit's digit of 7 in a positive integer power repeats in blocks of 4: 7-9-3-1. Since \(75=4*18+3\) then the unit's digit of \(7^{75}\) is the same as the unit's digit of \(7^3\), which is 3.

Therefore, the unit's digit of \(7^{75} + 6\) will be: 3 plus 6 = 9.


Answer: E


I don't understand that. I assumed that 75 is a multiple of 5, and 7^5 ends in 7, so I 7 + 6 = unit digit of 3. Where does that 4*18 + 3 come from? I mean.. I could simply say 75 = 9*8 + 3? Is that how you decide what unit digit it is?


1. 7^1=7 (units digit is 7)
2. 7^2=9 (units digit is 9)
3. 7^3=3 (units digit is 3)
4. 7^4=1 (units digit is 1)
5. 7^5=7 (units digit is 7 again!)
...

So, the units digit of 7^(positive integer) repeats in blocks of 4: {7, 9, 3, 1} - {7, 9, 3, 1} - ... (cyclicity of 7 in power is 4).

Now divide 75 (power) by 4 (cyclisity): the remainder is 3. So, the units digit of 7^75 is the same as that of the units digit of 7^3, which is 3.


Check Units digits, exponents, remainders problems directory in our Special Questions Directory.
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Re: M12-29 [#permalink] New post   23 Oct 2023, 11:20
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I've revised the question and solution, incorporating additional details for improved clarity. I trust this makes it more comprehensible.
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Re: M12-29 [#permalink]
23 Oct 2023, 11:20
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