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Bunuel
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The last digit of 12^12 + 13^13 – 14^14*15^15 = 09 Nov 2021, 03:00
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Difficulty:

95% (hard)

Question Stats:

37% (01:43) correct 63% (01:40) wrong based on 212 sessions

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The last digit of \(12^{12} + 13^{13} – 14^{14}*15^{15} =\)

A. 0
B. 1
C. 5
D. 8
E. 9
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B
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The last digit of 12^12 + 13^13 – 14^14*15^15 = 09 Nov 2021, 08:05
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Bunuel
The last digit of \(12^{12} + 13^{13} – 14^{14}*15^{15} =\)

A. 0
B. 1
C. 5
D. 8
E. 9
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\(14^{14}*15^{15}\) will surely end with 0, so we are looking only for units digit of \(12^{12} + 13^{13}\)

As \(14^{14}*15^{15}\) will surely be greater than \(12^{12} + 13^{13}\)

We can write \(-(14^{14}*15^{15}-12^{12} + 13^{13})\)

\(12^{12} + 13^{13}\) or \(12^{4*3} + 13^{4*3+1}\) will have same units digit as \(12^4+13^1\).
And \(12^{4} + 13^{1}\) will have same units digit as \(2^4+3^1\) or 16+3, that is 19.

Our answer is \(-(10-9)\) or 1.


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The last digit of 12^12 + 13^13 – 14^14*15^15 = 12 Apr 2023, 07:55
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Can someone please explain what the word last digit mean in this contex?

For example: what is the last digit in this number 34567890? is it 3 or 0?

Thanks in advance!
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The last digit of 12^12 + 13^13 – 14^14*15^15 = 12 Apr 2023, 08:00
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Rebaz
Can someone please explain what the word last digit mean in this contex?

For example: what is the last digit in this number 34567890? is it 3 or 0?

Thanks in advance!
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The last digit, refers to the units digit in a number. In your example, this digit would be 0.
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The last digit of 12^12 + 13^13 – 14^14*15^15 = 13 Apr 2023, 07:22
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Bunuel,

Thank you very much!
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The last digit of 12^12 + 13^13 – 14^14*15^15 = 13 Apr 2023, 09:51
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Bunuel
The last digit of \(12^{12} + 13^{13} – 14^{14}*15^{15} =\)

A. 0
B. 1
C. 5
D. 8
E. 9
Show more

It helps to think about this one in terms of cyclicity, which is a concept worth wrapping your head around, as it does show up on the test from time to time and knowing how it works makes questions like this MUCH easier.

The units digit for 2 (or any number with a 2 in the units digit) raised to powers cycles through 2, 4, 8, 6. The 12th term will be 6.
The units digit for 3 (or any number with a 3 in the units digit) raised to powers cycles through 3, 9, 7, 1. The 13th term will be 3.
The units digit for 4 (or any number with a 4 in the units digit) raised to powers cycles through 4, 6. The 14th term will be 6.
The units digit for 5 (or any number with a 5 in the units digit) raised to powers will always be 5. The 15th term will be 5.

6+3-6*5 = 9-30 = -21, which has a units digit of 1.

Answer choice E.
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