M ODULAR A RITHMETIC

1. MODULAR ARITHMETIC Spokane Math Circle May 5th 2012 DG Kim

2. Intro To Modular Arithmetic We usually think of numbers as they appear on a number line – stretching out infinitely in each direction…

3. Introduction Continued Modular arithmetic is thinking of numbers as repeating in cycles. Now we are going to develop a system of arithmetic around this kind of cyclical number system. Modular 7

4. Modular Arithmetic Examples 14 ≡ ? (mod 7) ? ≡ 0 (mod 7) 26 ≡ ? (mod 7) ? ≡ 5 (mod 7) 11 ≡ ? (mod 7) ? ≡ 4

5. Quick Quiz

6. 30 ≡ ? (mod 8) ? ≡ 6

7. 26 ≡ ? (mod 5) ? ≡ 1

8. 17 ≡ ? (mod 12) ? ≡ 5

9. 36 ≡ ? (mod 4) ? ≡ 0

10. 22 ≡ ? (mod 3) ? ≡ 1

11. Continuing…

12. Residues Quick definition: We say that r is the modulo m residue of n when n ≡ r (mod m) and 0 ≤ r < m. In that last section, all we were doing was calculating residues for given numbers and modulos. “Residue” is just the technical term.

13. Mod Arithmetic Properties

14. More Properties If:

15. Quick Quiz

16. 63 + 91 ≡ ? (mod 6) 3 + 1 ≡ 4 (mod 6)

17. 141 - 78 ≡ ? (mod 6) 3 - 0 ≡ 3 (mod 6)

18. 43 × 32 ≡ ? (mod 6) 1 × 2 ≡ 2 (mod 6)

19. 59 × 159 ≡ ? (mod 6) 5 × 3 ≡ 15 ≡ 3 (mod 6)

20. ≡ ? (mod 6) ≡ 64 ≡ 4 (mod 6)

21. Continuing…

22. Practical Usage Is a multiple of 11?

23. Solution We definitely don’t want to multiply out the exponentials, so we will use modular arithmetic. Now lets see how that applies in the equation: Since the residue results in 0 in mod 11, we can say that the exponents subtract to a multiple of 11.

24. Units Digit Mathematics Notice that the residues of anything in mod 10 is the units digit for the number: Similarly, the residues of anything in mod 100 will result in the last two digits of the number.

25. Example Find the units digit of 63 × 92. Solution Take the number in mod 10.

26. Example Find the last two digits of . Solution Take the number in mod 100:

27. Example Find the last two digits of Solution Take the number in mod 100.