- The number 2 divides both 18, and 84, but so does the number 3. So 6 will
also divide both numbers. Is there a larger number that will divide them? We
could try 9 or 18. On checking, these candidates do not divide 84 so 6 is the
largest number that divides both. We can conclude that 6 is the gcd of 18 and
84, writing this as gcd(18, 84) = 6.
The gcd can be interpreted in terms of kitchen tiling. It is the side of the
largest square tile that will tile a rectangular wall of breadth 18 and length 84,
with no cutting of tiles allowed. In this case, we can see that a 6 × 6 tile will do
the trick.
Tiling the square with a rectangular 18 × 84 tileThe greatest common divisor is also known as the ‘highest common factor’ or
‘highest common divisor’. There is also a related concept, the least common
multiple (lcm). The lcm of 18 and 84 is the smallest number divisible by both 18
and 84. The link between the lcm and gcd is highlighted by the fact that the lcm
of two numbers multiplied by their gcd is equal to the multiplication of the two
numbers themselves. Here lcm(18, 84) = 252 and we can check that 6 × 252 =
1512 = 18 × 84.
Geometrically, the lcm is the length of the side of the smallest square that can
be tiled by 18 × 84 rectangular tiles. Because lcm(a, b) = ab ÷ gcd(a, b), we’re