Reasoning · Grade 5-1 Greatest Common Divisor and Least Common Multiple
Problem
Both worked methods are loops.
Seeing the example as 'do this until you cannot any more' is what makes it reusable — otherwise it is just two pictures about 30 and 24.
6.NS.B.4Solve An Easier Related Problem72 and 34 share only one 2.
Writing both factorizations out in full turns 'what do they share?' into a matching game with no guessing left in it — a factor is shared only if it is physically in both lists.
6.NS.B.4Make A Systematic ListThe ladder also gives 2.
The stopping condition is the whole skill here: you do not stop when the numbers look small, you stop when the two quotients have nothing left in common.
6.NS.B.4Look For A Pattern36 and 54 share a 2 and two 3s.
'Take the smaller count of each prime' is exactly what matching does, and it explains why the answer can never be bigger than either number — you can only take what both actually have.
6.NS.B.4Make A Systematic ListMultiplying gives 18.
Each rung shrinks both numbers, so the ladder cannot run forever — it is guaranteed to reach a pair with nothing in common, and that is precisely when to stop.
4.NBT.B.6Look For A PatternThe ladder stops because the last quotients share nothing.
Reading the ladder backwards — rebuilding 36 and 54 from the leftovers — shows that the stopping condition is doing the 'greatest' part of the work, not just telling you when to put the pencil down.
6.NS.B.4Work BackwardsThe ladder has to stop, and the product of its rungs is the greatest common divisor.
Each rung makes both numbers strictly smaller, and numbers cannot keep shrinking below 1, so the loop cannot run forever.
Every rung peels off one shared prime, and the primes a number holds are fixed, so the rungs collect exactly the primes both numbers share.
The division chain also gives 2 and 18.
Three different loops, three different stopping conditions, one answer each time — which is the strongest possible sign that the answer belongs to the pair of numbers and not to the method.
6.NS.B.2Look For A PatternKeep peeling common factors off both numbers until nothing is shared any more — the stopping point is what makes your answer the GREATEST common divisor, not just a common one.