What number is the units digit of 357^110?

Question

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

OptimusPrepJanielle · Answer

357 = 3*7*17
357^110 = 3^110*7^110*17^110

To find the last digit, we need to know about the cyclicity of numbers.
Cyclicity = The power of a number after which the units digit is repeated

For example: 
2^1 = 2
2^2 = 4 
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64

Here the units digit is repeated after 2^4, hence cyclicity of 2 = 4

Cyclicity of 2, 3, 7, 8 = 4
Cyclicity of 4, 9 = 2
Cyclicity of 5, 6 = 1

Coming back to our question,
357^110 = 3^110*7^110*17^110
We know that 110 = 108 + 2 and the cyclicity of 3 and 7 = 4
Therefore 
Units digit of 3^110*7^110*17^110 is the same as units digit of 3^2*7^2*17^2
Units digit of 3^2*7^2*17^2 = 9*9*9 = 729

Hence the units digit of 357^110 = 9

Option E

stonecold · Answer

Cyclicity of 7 is 4 hence the UD is 9
Hit that E

cpat · Answer

Can you explain how you got this answer?

Is it that the cyclicity of 7 is 4 and 4*27=108 and 108+2 =110 so you go two more up in the cycle of 7 which is 9?

What number is the units digit of 357^110?

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What number is the units digit of 357^110?

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