Session 4C
Modular arithmetic
45-75 min - work through lesson notes, practice, and the MCQ check.
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Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- Modular arithmetic formalizes "same remainder." The statement a == b (mod m) means m divides (a - b), so a and b differ by a multiple of m. This definition is powerful because it turns a remainder claim into a divisibility claim, letting you use algebra safely. Once you know a == b (mod m), you can add the same number to both sides and keep congruence, subtract, and multiply, because these operations preserve the "difference is a multiple of m" property.
- The main beginner technique is reducing complicated expressions to small cases. For example, mod 2 there are only two residues: 0 and 1, corresponding to even and odd. Mod 4 there are residues 0,1,2,3, which lets you prove statements about squares by checking a small set of cases. Always be explicit about the modulus and what it means. Writing "mod m" without stating the definition is how mistakes happen. If you can restate congruence as "m | (a-b)" on demand, you are doing modular arithmetic correctly.
Practice
- 1Show: if a == b (mod m) then a + c == b + c (mod m).
- 2Compute: 17 mod 5 and 100 mod 7.
- 3Prove: if n is odd then n == 1 (mod 2).
MCQ
a == b (mod m) means: