What is the units digit of 4^93*9^15*3^81*7^56*2^9 ?

Question

What is the units digit of  4^{93}*9^{15}*3^{81}*7^{56}*2^9  ?

A. 0
B. 1
C. 4
D. 5
E. 6

Archit3110 · Answer

use cyclicity
we get 4*9*3*1*2 ; 216 
unit 6
IMO E

Lampard42 · Answer

Units digit of the bases are repeat in the following cycle - 4 for every 2nd time, 9 for every 2nd time, 3 for every 4th time, 7 for every 4th time, 2 for every 4th time.

the units digit of the expression will be - 4*9*3*1*2 whose unit digit will be 6 IMO option E

Archit3110 · Answer

Arvind42 
see highlighted part ; 9^15 ; shouldnt it be 1 ?

Lampard42 · Answer

Archit3110  9 repeats in a cycle of 2 i.e. 9 and 1 only. So for all odd powers units digit will be 9 and even powers 1. So i took 9.

Archit3110 · Answer

my bad ;; i did it right for others but got it wrong for 9 ..

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Arro44 · Answer

9^15 = 20589113209464 9

I also ended up at AC E with 6.

Abhishek009 · Answer

4^{93}*9^{15}*3^{81}*7^{56}*2^9

=  2^{186}*3^{30}*3^{81}*7^{56}*2^9

=  2^{195}*3^{111}*7^{56}

Now,  2^{195} = 2^{4*48+3}  ; So this will have units digit 8

3^{111} = 3^ {4*36 + 3}  will have units digit as 7

7^{56} = 7^{4*24}  will have units digit as 1

Finally we have units digit as 8*7*1 = xx56 , thus Answer must be (E) 6

fleecechief · Answer

Can you explain the cyclicality? howd you know to pick 3 instead of another number or the 1?

ScottTargetTestPrep · Answer

The pattern for a base of 4 is: 4-6-4-6, so we see that when 4 is raised to an odd power, the units digit is 4. Thus, the units digit of 4^93 is 4.

The pattern for a base of 9 is: 9-1-9-1, so we see that when 9 is raised to an odd power, the units digit is 9. Thus, the units digit of 9^15 is 9.

The pattern for a base of 3 is: 3-9-7-1-3-9-7-1, so we see that when 3 is raised to a power that is a multiple of 4, the units digit is 1. Thus, 3^80 has a units digit of 1, so 3^81 has a units digit of 3.

The pattern for a base of 7 is: 7-9-3-1-7-9-3-1, so we see that when 7 is raised to a power that is a multiple of 4, the units digit is 1. Thus, the units digit of 7^56 is 1.

Lastly, since 2^9 is a pretty small number, we see that 2^9 = 512, so it has a units digit of 2.

Since 4 x 9 x 3 x 1 x 2 = 216, the units digit of the original product is 6.

Answer: E

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