Module 3 · Number System
Chapter 35: Modular Arithmetic
Number System
3 topics
What this chapter covers
The 3 topics below make up Modular Arithmetic on the CAT. Click any topic to open its notes, examples and practice test.
Study material & tests
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Formula Reference Sheet
This chapter
Congruence basics & operations
| Definition | a ≡ b (mod m) ⇔ m divides (a − b) |
|---|---|
| Addition | a ≡ b, c ≡ d ⇒ a + c ≡ b + d (mod m) |
| Multiplication | a ≡ b, c ≡ d ⇒ ac ≡ bd (mod m) |
| Exponentiation | a ≡ b (mod m) ⇒ a^k ≡ b^k (mod m) |
| Reduce the base first | a^k mod m = (a mod m)^k mod m |
Inverses, theorems & calendars
| Modular inverse | a·a⁻¹ ≡ 1 (mod m); exists ⇔ gcd(a, m) = 1 |
|---|---|
| Linear congruence ax ≡ b | solvable ⇔ gcd(a,m) divides b |
| Fermat’s little theorem | p prime, gcd(a,p)=1 ⇒ a^(p−1) ≡ 1 (mod p) |
| Euler’s theorem | gcd(a,m)=1 ⇒ a^φ(m) ≡ 1 (mod m) |
| Day shift | days mod 7 → 0=same weekday, 1=next day, … |
CAT reference
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