GCD & LCM
Almost every inheritance calculation rests on two simple integer operations: LCM (Least Common Multiple) to unify fractions, and GCD (Greatest Common Divisor) to simplify them. Grasp both, and the "base of the problem" and "tashih" steps stop feeling mysterious.
- •The base of the problem (common denominator for all heirs) = the LCM of every fixed share’s denominator. Example: 1/2, 1/4, 1/6 unify on a base of 12.
- •Simplifying shares and tashih (ikhtisar an-nisab) use the GCD to keep numbers as small as possible before dividing among heads.
- •'Awl (shares exceed the base) and radd also come down to unifying denominators — again LCM and GCD.
GCD — Greatest Common Divisor
FPBThe largest number that divides two or more integers exactly. Used to simplify fractions and ratios.
Take the shared prime factors at the lowest power.
Replace the larger with the remainder, repeat until it reaches 0.
LCM — Least Common Multiple
KPKThe smallest number that is a multiple of two or more integers. Used to unify denominators — this is the base of the problem in faraidh.
Take all prime factors at the highest power.
LCM(a, b) = a × b ÷ GCD(a, b).
Learn with pictures
Three ways to see GCD & LCM so they are easier to grasp.
Keep splitting the number until every tip is a prime (gold circle). That is its factorisation.
Shared primes go in the middle (the overlap). From this one picture, both GCD and LCM appear at once.
Mark each number’s multiples on the line. The first meeting point is the LCM.
A man dies leaving a wife (1/8), his mother (1/6), and one son (the residue / 'asabah).
- 1. Fixed-share denominators: 8 and 6 → base = LCM(8, 6) = 24.
- 2. Shares: wife 1/8 × 24 = 3, mother 1/6 × 24 = 4, residue for the son = 24 − 7 = 17.
- 3. With a single son (no split across many heads) there is no tashih. When there are many heirs and the residue does not divide evenly, the GCD is used to simplify before tashih.
Try it in the Calculator — the base it shows is exactly the LCM of the denominators.