how to find the greatest common divisor of two numbers without factorising
Built as a memo rather than a table: the thing worth hanging it up for reads across the room, with the full reference in smaller type below it.
Three sizes in one archive — from a sheet on the fridge to a poster on the wall.
The GCD is the largest number that divides both exactly. The line “1071 = 2 × 462 + 147” reads: quotient 2, remainder 147.
The algorithm is slowest on neighbouring Fibonacci numbers, where each is the sum of the two before. 89 and 55 take 9 divisions: every quotient but the last is 1, and the remainders 34, 21, 13, 8, 5, 3, 2, 1 are Fibonacci numbers again. No pair whose larger number is below 89 needs that many. Hence Gabriel Lamé’s bound: never more than five divisions for each digit of the smaller number.
Why the numbers can be trusted. Every countable number here is computed from data rather than typed in, and the sheet must pass 3 independent checks before it is allowed to build — how many items there are, the order they come in, and the claim in the highlighted box itself. If one fails, there is no file.
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