Part 4/6:
One way to grasp modular arithmetic intuitively is by visualizing remainders around a circle. Imagine placing the number 9 at the top of a circle (the "modulus") and counting around:
- Starting at 9, moving forward by 4 steps lands you at 4, representing (13 \equiv 4 \pmod{9}).
This circular perspective is especially helpful for understanding how numbers "wrap around" when reaching the modulus.
Practical Application: Wrapping Numbers in Cycles
Suppose we want to determine what number 13 corresponds to in modulo 9:
13 divided by 9 leaves a remainder of 4.
On a clock or circular scale, starting at 9 and moving 4 steps forward lands you at 4.
Similarly, for larger numbers, this wrapping continues seamlessly:
- 17 in modulo 9 is 8 because (17 \div 9) leaves a remainder of 8.