Unit digit using modular arithmetic

[Sonal7]

Calculate 341+5679+4532+5991 (Mod 10).
What is the unit digit of 341+5679+4532+5991.

The first part is easy
341+5679+4532+5991(Mod 10) = 1+9+2+1 = 13 Mod 10 =3.

I can’t figure out unit digit from the remainder which I m sure is the idea.
i think the answer is 3 but the answer book says 2. I think I am right and the answer book is wrong !

 

[Romsek]

well checking it is pretty trivial... did you check it?
You're right, the units digit is 3.
The whole idea of this is that the units digit of the base 10 representation of (a bunch of stuff) is the same thing as (a bunch of stuff) (mod 10).

 

[Sonal7]

Okay. Another one whose answer is right as I checked using a calculator but I don’t get it I think.

consider the last digit of each of the following:
7, 7^2, 7^3, 7^4…7^6
find the last digit of 7^100.

well the pattern repeats 7,9,3,1,7,9…
100 mod 4 =0.
i think it should be 7. But the answer is 1 which has been checked using a calculator to be correct. 100 times of 7 multiplied by 7 should give you the digit 7???

ps... I am appending this 5 min later. I get it. It’s 7^0. It’s hard to explain but it’s the exponent that 7 needs to be raised by.

 

[Romsek]

I'm thinking you didn't read your textbook.

\(\displaystyle 7^{100} = ((7^4)^5)^5\\

7^{100} \pmod{10} =\left( \left(7^4 \pmod{10} \right)^5 \pmod{10}\right)^5 \pmod{10} = \\

1^{25} \pmod{10} = 1
\)

 

[Sonal7]

Thank you so much, I will keep trying to find similar examples from textbooks. The answer book says 7^0 =1. I guess there might be another way of doing things. Why rock the boat eh ? I think this is the right method. The answer book is confusing me but I need to look into the other method

 

[Deleted member 4993]

Thank you so much, I will keep trying to find similar examples from textbooks. The answer book says 7^0 =1. I guess there might be another way of doing things. Why rock the boat eh ? I think this is the right method. The answer book is confusing me but I need to look into the other method

The method is:

unit digit of 7¹ is 7 ............... 1 mod 4 = 1

unit digit of 7² is 9 ............... 2 mod 4 = 2

unit digit of 7³ is 3 ............... 3 mod 4 = 3

unit digit of 7⁴ is 1 ............... 4 mod 4 = 0 \(\displaystyle \ \ to \ \ \) unit digit of 7¹⁰⁰ is 1 ............... 100 mod 4 = 0