Remainders • Topic 2 of 3

Negative Remainders

A remainder need not be written as a positive number — and the cleverest CAT solutions use that freedom. If a number is just 1 short of a multiple of n, it is ≡ −1 (mod n); if 2 short, it is ≡ −2. The payoff is enormous for powers: when a base ≡ −1, any even power is +1 and any odd power is −1, so the whole exponent question collapses to a parity check. Example: the remainder of 6^100 mod 7 — since 6 ≡ −1 (mod 7), 6^100 ≡ (−1)^100 = 1. The discipline is two-step: first nudge the base to the nearest multiple of n to get −1 or −2, then read off the sign. When a calculation finishes with a negative remainder, convert to the standard positive form by adding n: a result of −3 (mod 7) is the usual remainder 4. CAT loves bases like 9 mod 10, 16 mod 17, and 22 mod 23 precisely for this −1 trick.

✅ Solved examples

1. Find the remainder when 6^100 is divided by 7.

6 ≡ −1 (mod 7). So 6^100 ≡ (−1)^100 = 1 (mod 7). Remainder = 1.

2. Find the remainder when 9^99 is divided by 10.

9 ≡ −1 (mod 10). 9^99 ≡ (−1)^99 = −1 ≡ 9 (mod 10). Remainder = 9.

3. Find the remainder when 22^63 is divided by 23.

22 ≡ −1 (mod 23). 22^63 ≡ (−1)^63 = −1 ≡ 22 (mod 23). Remainder = 22.

4. Find the remainder when 25^25 is divided by 26.

25 ≡ −1 (mod 26). 25^25 ≡ (−1)^25 = −1 ≡ 25 (mod 26). Remainder = 25.

✏️ Practice — try these, take hints as needed

1. Remainder when 8^40 is divided by 9.

8 ≡ −1 (mod 9).

(−1)^40 = +1.

Even exponent.

2. Remainder when 14^15 is divided by 15.

14 ≡ −1 (mod 15).

(−1)^15 = −1.

Convert −1 to positive: add 15.

3. Remainder when 4^53 is divided by 5.

4 ≡ −1 (mod 5).

(−1)^53 = −1.

Add 5 to get the standard remainder.

4. Remainder when 18^45 is divided by 19.

18 ≡ −1 (mod 19).

Odd power ⇒ −1.

−1 + 19.

5. Remainder when 5^30 is divided by 13.

5^2 = 25 ≡ −1 (mod 13).

So 5^30 = (5^2)^15 ≡ (−1)^15.

−1 ⇒ add 13.

📝 Topic test — 8 questions

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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