What does it mean when the GCD of two numbers is 1?

It means the two numbers share no divisor except 1, and they are called coprime, or relatively prime. 8 and 15 are a good example: 8 is 2 × 2 × 2 and 15 is 3 × 5, so no factor appears in both. Neither number has to be prime itself, since 8 and 15 are not.

How do you find the GCD of big numbers?

For big numbers, Euclid’s algorithm is faster than listing divisors. Divide the bigger number by the smaller one and keep the remainder. Then divide the smaller number by that remainder, and repeat until nothing is left. The last number you divided by is the GCD. For 165 and 132, the first remainder is 33, and 132 divided by 33 leaves 0, so the GCD is 33.

Is there a quick way to tell if two numbers share a divisor?

Yes, the divisibility rules give fast clues. When both numbers are even, 2 divides both. When both end in 0 or 5, 5 does. When the digits of each number add up to a multiple of 3, 3 divides both: for 84 and 141, the sums are 12 and 6. A shared clue proves a common divisor, but a bigger one may still be hiding.

Why does my child mix up GCD and LCM?

The two get mixed up because both take the same two numbers and both use the word “common”. The GCD divides both numbers, so it is never bigger than the smaller one. The LCM, or least common multiple, is the smallest number both divide into, so it is never smaller than the bigger one. For 6 and 8, the GCD is 2 and the LCM is 24.

Does this GCF game cost anything, and does it work for a class?

It costs nothing: both levels run in the browser on school laptops, tablets or phones, with no sign-ups or installs. For highest common factor practice at the board, draw two overlapping circles. Pupils fill one with the divisors of 18 and the other with those of 24, then move the shared ones, 1, 2, 3 and 6, into the overlap. The largest, 6, is the GCD. The schools plan, priced per student, adds a teacher dashboard of who is improving.

Math practice · reimagined

GCD Game

This greatest common divisor game shows two numbers between 4 and 200 and asks for the biggest number that divides both, which can be as small as 1. Find it while your rival is still searching, and your fighter swings first.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is the greatest common divisor?

The greatest common divisor, or GCD, of two numbers is the biggest number that divides both with nothing left over. It is also called the greatest common factor (GCF) or highest common factor (HCF). Share 8 strawberries and 12 grapes between plates so every plate looks the same: 4 plates is the most, with 2 strawberries and 3 grapes each.

Every level of GCD

Example: find GCD(30, 24)

The screen shows GCD(30, 24), so you want the biggest number that divides both 30 and 24.
List the divisors of the smaller number, 24: they are 1, 2, 3, 4, 6, 8, 12 and 24.
Test them on 30, starting from the top: 24, 12 and 8 leave a remainder, but 30 ÷ 6 = 5 exactly.
So the GCD is 6, and you type 6 on the keypad.

Learn it step by step

GCD means "greatest common divisor" — the biggest number that divides both numbers cleanly, with no remainder.

For 12 and 18: numbers that divide both are 1, 2, 3, 6 — and the biggest is 6. So gcd(12, 18) = 6. Think of the largest tile that could pave two different rooms exactly.

Type the greatest shared divisor.

Tips and common mistakes

Read the name from the end and it gives you the method: list each number’s divisors, keep the common ones, then take the greatest.
A common mistake is stopping at the first shared divisor. 18 and 42 are both even, but 6 divides both as well, so GCD(18, 42) is 6, not 2.
Take two family ages, such as 12 and 40, and race to find the biggest number dividing both. It is 4: 12 is three 4s, and 40 is ten 4s.

How to play

Every round shows a pair such as GCD(26, 4), and you type the largest number that divides both, here 2, on the keypad. Some pairs share nothing but 1, like GCD(26, 17), so 1 is a real answer. Your rival works on the same pair: answer correctly first and Tomato strikes, with speed adding weight to the blow.

— Quick answers —

GCD: the short FAQ

What does it mean when the GCD of two numbers is 1?: It means the two numbers share no divisor except 1, and they are called coprime, or relatively prime. 8 and 15 are a good example: 8 is 2 × 2 × 2 and 15 is 3 × 5, so no factor appears in both. Neither number has to be prime itself, since 8 and 15 are not.
How do you find the GCD of big numbers?: For big numbers, Euclid’s algorithm is faster than listing divisors. Divide the bigger number by the smaller one and keep the remainder. Then divide the smaller number by that remainder, and repeat until nothing is left. The last number you divided by is the GCD. For 165 and 132, the first remainder is 33, and 132 divided by 33 leaves 0, so the GCD is 33.
Is there a quick way to tell if two numbers share a divisor?: Yes, the divisibility rules give fast clues. When both numbers are even, 2 divides both. When both end in 0 or 5, 5 does. When the digits of each number add up to a multiple of 3, 3 divides both: for 84 and 141, the sums are 12 and 6. A shared clue proves a common divisor, but a bigger one may still be hiding.
Why does my child mix up GCD and LCM?: The two get mixed up because both take the same two numbers and both use the word “common”. The GCD divides both numbers, so it is never bigger than the smaller one. The LCM, or least common multiple, is the smallest number both divide into, so it is never smaller than the bigger one. For 6 and 8, the GCD is 2 and the LCM is 24.
Does this GCF game cost anything, and does it work for a class?: It costs nothing: both levels run in the browser on school laptops, tablets or phones, with no sign-ups or installs. For highest common factor practice at the board, draw two overlapping circles. Pupils fill one with the divisors of 18 and the other with those of 24, then move the shared ones, 1, 2, 3 and 6, into the overlap. The largest, 6, is the GCD. The schools plan, priced per student, adds a teacher dashboard of who is improving.

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