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What is the units digit of 6^15 - 7^4 - 9^3?

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by Anurag@Gurome » Mon Jul 09, 2012 3:04 am
gmatter2012 wrote:What is the units digit of 6^15 - 7^4 - 9^3?
Any power of 6 always ends with 6.
Now, unit's digit of 7^4 = 1
And, unit's digit of 9^3 = 9

Hence, unit's digit of (7^4 + 9^3) = unit's digit of (1 + 9) = 0

Therefore, unit's digit of (6^15 - 7^4 - 9^3) = unit's digit of (6 - 0) = 6

The correct answer is C.
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by gmatter2012 » Mon Jul 09, 2012 3:19 am
Anurag@Gurome wrote:
gmatter2012 wrote:What is the units digit of 6^15 - 7^4 - 9^3?
Any power of 6 always ends with 6.
Now, unit's digit of 7^4 = 1
And, unit's digit of 9^3 = 9

Hence, unit's digit of (7^4 + 9^3) = unit's digit of (1 + 9) = 0

why are we adding here?

Therefore, unit's digit of (6^15 - 7^4 - 9^3) = unit's digit of (6 - 0) = 6

The correct answer is C.
What I would do is (6 - 1 - 9 )= -4 which is not the answer

what I cannot understand is why are we adding units digit of 7^4 + 9^3 shouldn't we subtract?
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by Anurag@Gurome » Mon Jul 09, 2012 3:37 am
gmatter2012 wrote:What I would do is (6 - 1 - 9 )= -4 which is not the answer

what I cannot understand is why are we adding units digit of 7^4 + 9^3 shouldn't we subtract?
You can either add first then subtract or you can directly subtract.
The order of addition and subtraction doesn't matter what you do as long as you are doing it in proper way.

Now what is your logic of saying -4 is the unit's digit?
Have you ever seen any number whose unit's digit is negative? Or any other digit for that matter?

What is the unit's digit of (66 - 21 - 19)?
Is it -4? Or it is the unit's digit of 26?

(66 - 21 - 19) will always be equal to 26.
It does not mater how you get 26. You can get 26 in any of the following ways...
  • (66 - 21 - 19) = (66 - (21 + 19)) = (66 - 40) = 26
    (66 - 21 - 19) = ((66 - 21) - 19) = (45 - 19) = 26
    (66 - 21 - 19) = ((66 - 19) - 21) = (47 - 21) = 26
Similarly unit's digit of (some integer with unit's digit 6 - some integer with unit's digit 1 - some integer with unit's digit 9) = 6
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by gmatter2012 » Mon Jul 09, 2012 4:12 am
Anurag@Gurome wrote:
gmatter2012 wrote:What I would do is (6 - 1 - 9 )= -4 which is not the answer

what I cannot understand is why are we adding units digit of 7^4 + 9^3 shouldn't we subtract?
You can either add first then subtract or you can directly subtract.
The order of addition and subtraction doesn't matter what you do as long as you are doing it in proper way.

Now what is your logic of saying -4 is the unit's digit?
Have you ever seen any number whose unit's digit is negative? Or any other digit for that matter?

What is the unit's digit of (66 - 21 - 19)?
Is it -4? Or it is the unit's digit of 26?



(66 - 21 - 19) will always be equal to 26.
It does not mater how you get 26. You can get 26 in any of the following ways...
  • (66 - 21 - 19) = (66 - (21 + 19)) = (66 - 40) = 26
    (66 - 21 - 19) = ((66 - 21) - 19) = (45 - 19) = 26
    (66 - 21 - 19) = ((66 - 19) - 21) = (47 - 21) = 26
Similarly unit's digit of (some integer with unit's digit 6 - some integer with unit's digit 1 - some integer with unit's digit 9) = 6

Thank you Anurag But I am obviously missing something Please try and figure out what I am missing so that we can be on the same page( Request you please bear with me and maintain your cool )
If I was given the sum (66 - 21 - 19) and asked to calculate the value of this operation I would get
26 because I would add the negative terms and subtract from 66 so I would be adding -19 and -21 which gives 40 and subtracting 40 from 66 gives 26 as [ 66-(21 + 19)]= [60-21-19]

if someone asked me the units digit of (66 - 21 - 19)I would say 6 ( Not 26)

Now given 6^15 - 7^4 -9^3
6^15 units digit 6 [I don't know how many digits 6^15 has, I only know the units digit ]
7^4 units digit 1 [I don't know how many digits 7^4 has, I only know the units digit ]
9^3 units digit 9 [I don't know how many digits 9^4 has, I only know the units digit ]

so when finding the answer of this operation (6-1-9) what would I do?
I would do as I did above I would add the negative terms and subtract from the positive terms
6-10 = -4 now this would be the answer in a normal scenario

but over here this is not the answer as we are not finding the value of the expression (6-1-9)but rather the unit digit of the expression (6-1-9)so here is what I am missing

This is what I need someone to explain to me? what is the concept of adding or subtracting when finding the units digit ?

My own immature temporary concept : Follow normal algebraic rules with the unit digits and if the units digit comes out negative then subtract from 10 . obviously I need to understand the concept rather than mug it up.

Kindly bear with me
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by Anurag@Gurome » Mon Jul 09, 2012 4:26 am
My own immature temporary concept : Follow normal algebraic rules with the unit digits and if the units digit comes out negative then subtract from 10.
What you have said is correct except that it is not temporary concept. It is how subtraction is carried out. I've shown you the example with 66, 21, and 19. 6 is less than 1 and 9 put together. But we have to subtract 1 and 9 from 6. Hence, we borrow 1 from 6 (the tens digit of 66) and add 10 to 6 (unit's digit of 66). Now our job is to subtract 1 and 9 from (10 + 6) = 16. And we put (16 - 1 - 9) = 6 in the unit's place.

Similar will be the case when we perform the operation (some integer with unit's digit 6 - some integer with unit's digit 1 - some integer with unit's digit 9). We will borrow one from the tens digit of some integer with unit's digit 6 and add 10 to the unit's digit, i.e. 6. And subtract 1 and 9 put together from (10 + 6) = 16. And similarly we will put (16 - 1 - 9) = 6 in the unit's place.

Hope that helps.

Note: The operations (some integer with unit's digit 6 - some integer with unit's digit 1 - some integer with unit's digit 9) and the unit's digit of (some integer with unit's digit 6 - some integer with unit's digit 1 - some integer with unit's digit 9) are NOT same.

For example,
  • 1. (66 - 21 - 19) = 26 but unit's digit of (66 - 21 - 19) = 6
    2. (26 - 21 - 19) = -14 but unit's digit of (26 - 21 - 19) = 4
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by gmatter2012 » Mon Jul 09, 2012 4:46 am
Bingo!! You directly hit the Bull's eye there

This was what I was missing , the carry over funda.

I was not visualizing the carry over concept, of course 6 becomes 16 and then we subtract 10 from 16 to get 6.

Wish I would Thank you 2 times for this !!

Great!
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