(a+b) mod n = (a mod n + b mod n) mod n
(a×b) mod n = (a mod n × b mod n) mod n
DIVISION DOES NOT WORK THIS WAY.

⭐ NEGATIVE REMAINDER: if a ≡ −1 (mod n)
     even power → 1        odd power → n−1
   (6^100 mod 7 = 1, instantly)

⭐ FERMAT (divisor PRIME p):   a^(p−1) ≡ 1 (mod p)
     3^100 mod 7: 3^6≡1, 100 mod 6 = 4, → 3^4 = 81 ≡ 4

⭐ EULER (divisor COMPOSITE, coprime): a^φ(n) ≡ 1 (mod n)
     φ(n) = n(1−1/p₁)(1−1/p₂)…
     φ(10)=4  φ(100)=40  φ(25)=20  φ(36)=12  φ(prime p)=p−1

CYCLICITY: list powers until they repeat; take exponent mod cycle length.
   If exponent divides evenly → use the LAST element of the cycle.

SAME REMAINDER for a,b,c → HCF of the DIFFERENCES
SAME REMAINDER r by x,y,z → LCM(x,y,z) + r
CONSTANT GAP (div−rem equal) → LCM − gap

Two-digit number = 10a + b
  reversed = 10b + a
  ⭐ difference = 9(a−b)   ·  sum = 11(a+b)

UNIT DIGIT: exponent mod 4
  0,1,5,6 → itself   ·  4,9 → cycle 2   ·  2,3,7,8 → cycle 4
  2: 2,4,8,6   3: 3,9,7,1   7: 7,9,3,1   8: 8,4,2,6
  If exp mod 4 = 0 → last element of cycle

⭐ TRAILING ZEROS in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ …   (count 5s, never 2s)

N = pᵃqᵇrᶜ  →  factors = (a+1)(b+1)(c+1)
   ODD factors: drop the power of 2 entirely

HCF × LCM = product of the two numbers
   largest tile / greatest measure → HCF
   bells together again / smallest divisible → LCM

RECURSION f(2n)=f(n), f(2n+1)=f(n)+1 → counts the 1s in binary

⭐ km/h → m/s: × 5/18     m/s → km/h: × 18/5   (DO THIS FIRST, ALWAYS)

TRAINS  pole → train length
        platform → train + platform
        another train → sum of both lengths
RELATIVE  opposite → add    ·   same direction → subtract

BOATS   down = b+s   up = b−s
        b = (down+up)/2      s = (down−up)/2

⭐ AVERAGE SPEED (equal DISTANCES) = 2ab/(a+b)   [harmonic — never (a+b)/2]
   (equal TIMES → then it IS (a+b)/2)

⭐ Same distance → speeds a:b means times b:a

FRACTIONS  1/2=50  1/3=33.33  1/4=25  1/5=20  1/6=16.67  1/7=14.28
           1/8=12.5  1/9=11.11  1/11=9.09  1/12=8.33  3/8=37.5  5/8=62.5

⭐⭐ MULTIPLIER METHOD (use this by default — handles ANY number of changes)
   +20% → 1.20   −15% → 0.85   +40% → 1.40   −25% → 0.75
   +40% then −25%      → 1.40 × 0.75 = 1.05      → net +5%
   +10%, −20%, +25%    → 1.10 × 0.80 × 1.25 = 1.10 → net +10%
   (+20% then −20% → 1.20 × 0.80 = 0.96 → the 4% loss is VISIBLE)

⭐ SUCCESSIVE a then b = a + b + ab/100      (two changes only)
⭐ Up x% then down x% = net LOSS of x²/100 %

REVERSE  A is x% more than B → B is [x/(100+x)]×100 % less than A
         A is x% less than B → B is [x/(100−x)]×100 % more than A
   (price up 25% → need 20% fall to return)

Price up x% → consumption down x/(100+x) × 100 %  (expenditure fixed)