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Arithmetic 07 Aug 2026, 10:33
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ARITHMETIC — SAVED LEARNINGS ONLY
  1. SUM OF FIRST n INTEGERS
1 + 2 + 3 + ... + n = n(n + 1)/2
  1. SUM OF FIRST n ODD NUMBERS
1 + 3 + 5 + ... = n2
  1. SUM OF FIRST n EVEN NUMBERS
2 + 4 + 6 + ... + 2n = n(n + 1)
  1. SAME REMAINDER ⭐
If A and B leave the same remainder when divided by d:
d divides (A - B)
Recognition:
Same remainder → SUBTRACT.
  1. REMAINDER FORM
N = dq + r
where:
0 ≤ r < d
Important: The remainder must always be from 0 to d - 1.
  1. POLYNOMIAL REMAINDER
If:
N ≡ r (mod d)
Then:
P(N) ≡ P(r) (mod d)
Recognition:
Polynomial → replace the number by its remainder.
  1. REMAINDER OPERATIONS
Replace numbers by their remainders as early as possible.
Then REDUCE AFTER EVERY OPERATION.
Important: Never calculate huge numbers if you only need the remainder.
  1. FACTOR BEFORE FINDING CYCLES ⭐
Before looking for a repeating remainder cycle, check whether the expression can be factored.
Example:
2100 + 299
= 299(2 + 1)
= 3 × 299
Therefore, it is divisible by 3.
Recognition:
Powers → FACTOR FIRST → then check cycles only if necessary.
  1. NEGATIVE REMAINDER
If you get a negative remainder, keep adding the divisor until:
0 ≤ r < d
Example:
-7 mod 13
-7 + 13 = 6
Therefore:
-7 mod 13 = 6
  1. FACTORIALS & DIVISIBILITY
If n! contains all the prime factors of divisor d, including the required number of copies of each prime factor, then:
n! ≡ 0 (mod d)
In other words:
d divides n!
  1. CONSECUTIVE INTEGERS ⭐
The product of k consecutive integers is divisible by:
k!
Example:
n(n + 1)(n + 2)(n + 3)
is divisible by:
4! = 24
Recognition:
Product of k consecutive integers → divisible by k!
  1. DIFFERENCE OF POWERS
an - bn is divisible by:
a - b
Example:
17100 - 5100
is divisible by:
17 - 5 = 12
Recognition:
an - bn → think a - b.
  1. DIVISIBILITY — LINEAR COMBINATION
If:
a divides x
and:
a divides y
Then:
a divides (mx + ny)
for integers m and n.
Recognition:
If the same number divides two quantities, it also divides any integer linear combination of them.
  1. FACTOR PAIR SYMMETRY
Factors of N occur in pairs around:
√N
Example:
For N = 36:
1 × 36
2 × 18
3 × 12
4 × 9
6 × 6
Recognition:
To find factor pairs, you only need to check up to √N.
  1. SUM OF POSITIVE FACTORS
If:
N = p1^a1 × p2^a2 × ... × pₖ^aₖ
Then the sum of all positive factors is:
(1 + p1 + p12 + ... + p1^a1)
×
(1 + p2 + p22 + ... + p2^a2)
×
...
×
(1 + pₖ + pₖ2 + ... + pₖ^aₖ)
  1. PRODUCT OF ALL POSITIVE FACTORS
If N has d positive factors:
Product of all positive factors = N^(d/2)
Recognition:
Pair each factor with N/factor → every pair has product N.
  1. COMMON MULTIPLES
Common multiples of A and B are multiples of:
LCM(A, B)
To count the number of common multiples up to N:
⌊N / LCM(A, B)⌋
Recognition:
"Divisible by BOTH A and B" → think LCM.
  1. SUM OF NUMBERS FORMED — NO REPETITION
Using n digits exactly once:
Sum of all numbers formed
= (n - 1)! × (sum of digits) × (111...1)
where 111...1 contains n ones.
Reason:
Each digit appears (n - 1)! times in every position.
  1. SUM OF NUMBERS FORMED — REPETITION ALLOWED
If there are k available digits and n positions:
Sum of all numbers formed
= k^(n - 1) × (sum of digits) × (111...1)
where 111...1 contains n ones.
Reason:
Each available digit appears k^(n - 1) times in every position.
  1. POWERS OF 10 — NUMBER OF DIGITS
10n has:
n + 1 digits
Examples:
102 = 100 → 3 digits
105 = 100000 → 6 digits
  1. REMAINDER MASTER RECOGNITION ⭐
When you see:
• Same remainder → SUBTRACT.
• Polynomial → SUBSTITUTE THE REMAINDER.
• Powers → FACTOR BEFORE LOOKING FOR CYCLES.
• Huge numbers → NEVER CALCULATE THEM DIRECTLY; REDUCE EARLY.
• Negative remainder → ADD THE DIVISOR until 0 ≤ r < d.
• Product of k consecutive integers → divisible by k!.
• an - bn → divisible by a - b.
• Divisible by BOTH A and B → think LCM.
• Factor questions → factors pair around √N.
REMAINDERS — MASTER RECOGNITION RULES ⭐
  1. Never think in words. Immediately write:
N = (Divisor × Integer) + Remainder
  1. Never calculate huge numbers. Replace every number or variable by its remainder as early as possible.
  2. Whenever you see exponents, check for repeating remainder (modular) cycles.
  3. Before finding cycles, always check whether the expression can be FACTORED.
  4. If two numbers leave the SAME remainder:
Subtract them.
  1. If you have a polynomial expression:
Replace the input by its remainder.
  1. Reduce after EVERY operation.
  2. The final remainder must satisfy:
0 ≤ remainder < divisor.
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Math Expert
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