GCD Ladder Game

How to Play GCD Ladder

Find the greatest common divisor of the displayed numbers. This is also called the greatest common factor, or GCF. Begin at the secured divisor 1 and choose a higher candidate rung. A move is legal only when that candidate divides every displayed number exactly. A divisor belonging to just one number cannot be used to climb this ladder.

Each accepted rung shows the exact division results for all the numbers. A rejected rung explains a nonzero remainder, so you can see why it is not shared. You may jump directly to a higher legal rung rather than visiting every intermediate divisor. Once you believe the secured divisor is the greatest, select Check greatest divisor.

A shared divisor may still be too small. In that case the check tells you to keep climbing without treating that rung as the final answer. The candidate list contains the greatest common divisor as well as distractors. Your objective is the largest mathematically valid shared divisor, rather than simply choosing the largest number visible on the board.

Worked Example: GCD of 18 and 24

The positive divisors of 18 are 1, 2, 3, 6, 9 and 18. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Their shared divisors are 1, 2, 3 and 6, so the greatest common divisor is 6. The divisions 18 ÷ 6 = 3 and 24 ÷ 6 = 4 both leave no remainder.

A proposed rung of 9 fails because 24 ÷ 9 leaves remainder 6, even though 9 is a divisor of 18. A rung of 3 is shared, but it is not the greatest because 6 is also shared. This distinction is the reason the game separates a legal climb from checking the final greatest divisor.

After completion, open the Euclidean algorithm check to review another proof. For 24 and 18, write 24 = 1 × 18 + 6, then 18 = 3 × 6 + 0. The last nonzero remainder is the GCD, 6. For three numbers, take the GCD of the first two and then combine that result with the third. If the only common divisor is 1, the numbers are relatively prime.

Levels, Common Factors and Saved Puzzles

Easy uses two smaller numbers and can include relatively prime pairs. Normal uses two larger numbers with additional candidate factors. Hard asks for a divisor shared by three numbers, requiring you to check every value. A candidate that fits two out of three numbers is still rejected. All choices use whole-number arithmetic and exact remainders.

Tap a higher rung or choose its button with Tab and Enter or Space. The current and lower rungs stay secured instead of allowing downward movement. Undo returns to your previous secured divisor and marks assistance. A hint proposes a higher shared candidate, or tells you when there is no higher common divisor left.

Daily mode provides a repeatable puzzle for each UTC date and level, while Practice offers another set through New puzzle. Your ladder, result and records save locally when available. Wrong choices reduce the score by five points and hints by fifteen, with a ten-point floor on completion. Hints and Undo mark assistance; assisted rounds do not replace an unaided best, and replaying the same puzzle does not add another completion.

Frequently Asked Questions

Are GCD and GCF the same?

Yes. Greatest common divisor and greatest common factor mean the largest positive whole number that divides every supplied number.

Can the answer be 1?

Yes. If no larger positive divisor is shared, the GCD is 1.

How is the GCD of three numbers checked?

Find the GCD of the first two numbers, then find the GCD of that result and the third number.