Unit digit-how to find out?

[Mahi111]

Can someone please tell me how to find of the unit digit of the following.

Find out the unit digit of 4^27 * 5^27 * 3^27?

Not only the above one,how to find unit digit of any value to any power?

Please some one explain me this concept.

Thx.

[moe]

5, 15, 25.... to any power, always unit digit=5, and same goes for 6

others follow easy patterns:

the easiest are 4 and 9:

4, 6, 4, 6.... and 9, 1...

2^1=2
2^2=4
2^3=8
2^4=16
2^5=32 (back to the start, unit = 2)

for 8 it's simmilar to 2:
8, 4, 2, 6

3 and 7 are simmilar too:

3, 9, 7, 1 and 7, 9, 3, 1

helps?!

[moe]

and to find out the unit digit of 4^27 * 5^27 * 3^27... 4^27 will have unit = 4 and 5^27, unit = 5 ==> 4^27 * 5^27, unit = 0 ==> 4^27 * 5^27 * 3^27, unit = 0

is that the answ?

[Mahi111]

Thank you moe,Yes.u r answer is correct.

but I did not understand this part

and anything to the power of 2, mean that first I should check where it is repeating,then, that reaping one is the unit digit or else ,I strucked up here.


2^1=2
2^2=4
2^3=8
2^4=16
2^5=32 (back to the start, unit = 2)


for 8 it's simmilar to 2:
8, 4, 2, 6

3 and 7 are simmilar too:

3, 9, 7, 1 and 7, 9, 3, 1

I think I have not understand the concept perfectly,can you please help me by saying bit clearly.

[moe]

ok... lets see if i explain myself any better...

2^1=2
2^2=4
2^3=8
2^4=16
2^5=32 ==> same unit digit as 2 to the power of 1 (2^1=2)
2^6=64 ==> same unit digit as 2 to the power of 2 (2^2=4)
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..
...
units digit can be grouped into a sequence of 4 numbers that repeat themselves 2/4/8/6==> 2 will be the unit number for 2 to the power of 1, 5, 9, 13... and, in the same way, 4 will be the unit number for 2 to the power of 2, 6, 10, 14...
eg: what's the unit digit of 2^16? 16/4=4 no remainder, therefore, the units digit will be the last one in the series=6

(and this same sequence applies to the powers of numbers 12, 22, 32... 32^3 will have a unit digit of 8, the same as 22^3 and 2^3)

numbers 3, 7 and 8 also have sequences of 4 numbers:
for nº3: 3/9/7/1
eg:3^4 will have a unit digit of 1 (and 13^4 as well), 3^5 will have unit digit 3...
the sequences of 7 (7, 9, 3, 1) and 8 (8, 4, 2, 6) follow the same rules

[sruthi07]

explanation moe, and time saving too!!
thank you!

i had a diff approach, let me know whther im right

viz; a ^b * c^d * e ^f ; unit digit of this one will be

a^1* b^1* c^1...???

so in the problem above
5*4*3 has a unit digit = 0

[moe]

i'm sorry, but i'm not sure i get the method... if i understand well, in a shorter example, like 4^2*3^7, u would do 4*2*3*7? that wouldn't work on this new example...

[gre_girl]

hi moe
sory but i cant understand this concept.can u explain again
thanks in advance
regards

[whaw]

helo mahi111, u can solv for the units digit by getting the patern...here's how

4^1=4
4^2=6
4^3=4
4^4=6

So 27/2=13 remainder 1 therefore units digit is 4.

5^1=5
5^2=5
5^3=5...therefore the units digit is 5...and so on..

[taminkeu]

4^27 * 5^27 * 3^27 = (4*5*3)^27 = 60^27
unit of 60^1 = 0
unit of 60^2 = 0
unit of 60^3 = 0
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unit of 60^27= 0