Learning pathOriginal reference
Lesson 2 of 8

Combine periods with a safe LCM

Divide first and guard the product that remains.

Understand the idea

For positive a and b, lcm(a,b) = (a/gcd(a,b)) × b. GCD takes minimum prime exponents and LCM takes maxima. Dividing first avoids an unnecessarily large intermediate product, but the final result still must fit.

Trace a small example

Periods 12 and 18 coincide every 36 units. For cap L and positive b, set q = a/gcd(a,b); if q > L/b, the LCM exceeds L without calculating an unsafe product. On a circle, n/gcd(n,k) moves of size k return to the start.

Conditions and cost

Handle zero inputs before division: the usual nonnegative LCM convention returns zero if either input is zero. Circle formulas require n > 0. Signed input normalization and final-product bounds are separate obligations.

Pause & predict

Does dividing by the GCD guarantee the LCM fits?

Choose an answer
Try it yourself

Turn the idea into code

On a circle of 12 positions, how many moves of size 8 return to the start?

Give me a hint

Find n/gcd(n,k).

Show the explained solution

12/4 = 3 moves: 0 → 8 → 4 → 0.

Your notes

Ready for the next idea?

Mark complete when you can explain it yourself.

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