Module 3 · Number System
Chapter 34: Remainders
Number System
3 topics
What this chapter covers
The 3 topics below make up Remainders on the CAT. Click any topic to open its notes, examples and practice test.
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Formula Reference Sheet
This chapter
Modular reduction core
| Notation | a ≡ r (mod n) means n divides (a − r) |
|---|---|
| Sum / difference | (a ± b) mod n = (a mod n ± b mod n) mod n |
| Product | (a × b) mod n = ((a mod n)(b mod n)) mod n |
| Power | a^k mod n = (a mod n)^k mod n |
| Negative remainder | if a ≡ −1 (mod n) then a^k ≡ (−1)^k (mod n) |
Named theorems
| Fermat’s little theorem | a^(p−1) ≡ 1 (mod p), p prime, gcd(a,p)=1 |
|---|---|
| Euler’s theorem | a^φ(n) ≡ 1 (mod n), gcd(a,n)=1 |
| Euler’s totient | φ(n) = n·∏(1 − 1/p) over distinct primes p | n |
| Wilson’s theorem | (p − 1)! ≡ −1 (mod p), p prime |
| Corollary of Wilson | (p − 2)! ≡ 1 (mod p), p prime |
CAT reference
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