1. anonymous

$m= 10^{32}+2$ When m is divided by 11, the remainder is r Which one is greater or are they equal. Quantity A: r Quantity B: 3

2. freckles

$10^2 \mod 11 =100-9(11)=1 \\ 10^{2 \cdot 16 } \mod 11 \equiv (10^2)^{16} \mod 11 \equiv ...$ I'm just using the repeated squares method to find what 10^(32) divided by 11 is

3. freckles

do you think you can finish this method

4. anonymous

huh?

5. anonymous

what is mod?

6. anonymous

7. freckles

well basically a mod n gives us the remainder for when a is divided by n

8. freckles

but there are lots of method to help us solve the above but if you don't know mod then it likely you don't know methods such as the repeated squares method

9. freckles

ok so you want to see if you can a pattern in the remainders above well 10+2 is 12 what is the remainder of 12 divided by 11 ?

10. anonymous

1

11. freckles

|dw:1437432052883:dw| ok let's continue what is the remainder of 10^4+1 when it is divided by 11

12. anonymous

OHHHHHH okay so it'll be 3

13. freckles

|dw:1437432204940:dw| so the pattern seems to be alternating between 1 and 3

14. freckles

so notice the odd exponents on the 10 gave us 1 and the even exponents on the 10 gave us 3

15. freckles

32 is even

16. anonymous

yassssssssssss

17. freckles

so 10^(32)+2 mod 11 is 3 and since you don't know mod $\text{ remainder of } \frac{10^{32}+2}{11} \text{ is } 3$