- Middle school๐ขArithmeticโ
Simplify a square root: hunting for perfect squares
โ72 = 6โ2 in 5 seconds. The method that pulls out perfect squares without breaking a sweat.
- Middle school๐ขArithmeticยฑ
Sign rules: + ร โ = โ, the table to memorize
Multiplying or dividing signed numbers: two rules cover it. For addition/subtraction, the method that avoids every trap.
- Middle school๐ขArithmetic๐ณ
Prime factorization: the tree method
Break a number into prime factors. Useful to simplify fractions, compute GCD and LCM.
- Middle school๐ขArithmetic๐งฎ
Powers of 2 to memorize (up to 1024)
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024โฆ A short list that unlocks tons of mental math โ and all of computing.
- Middle school๐ขArithmetic๐งฉ
Find the GCD without a single division
Euclid's algorithm, mental version: subtract the small from the big until you get the same value. That's the GCD.
- Middle school๐ขArithmetic๐ข
Euclidean algorithm: GCD by successive divisions
A systematic method to find the GCD of two integers. Successive divisions, stop when the remainder hits 0.
- Middle school๐ขArithmetic๐
Divisibility by 3 and 9: add up the digits
A number is divisible by 3 (or 9) if the sum of its digits is. Mind-blowing test, works for all integers.
- Middle school๐ขArithmeticโ๏ธ
Are two fractions equal? The cross-product decides
$\dfrac{a}{b} = \dfrac{c}{d}$ if and only if $ad = bc$. A foolproof test, no common denominator needed.
- High school๐ขArithmetic๐
Congruences modulo n: a โก b [n] if a and b share a remainder
17 โก 5 [12] since 17 = 12 + 5. A cyclic arithmetic simplifying divisibility and cryptography.
- High school๐ขArithmetic๐ค
Bรฉzout's identity: au + bv = gcd(a, b)
There exist integers u, v with au + bv = gcd(a,b). Found by extended Euclidean algorithm.
- Middle school๐ขArithmeticโ
Adding two fractions: the common denominator without hunting
$\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$. A simple formula that skips the search for the least common multiple.