I was given a question to find a greatest possible length to measure $495$, $900$,$1665$ (in centimetres).
The solution is finding GCF or GCD or HCF (highest common factor) of these numbers which is $45$.
How does this works in real time, please explain in layman words
Thank you all
The way that we measure lengths is to have a standard unit of measurement and then count how many of these units a given length is. Common standard units include millimetre, centimetre, metre, kilometre, inch, foot, mile, lightyear, astronomical unit, etc.
Example: An Olympic size swimming pool is $50$ metres long; this means that if I had $50$ one-metre long pieces of wood, I could lay them end to end and they would stretch from one end of the pool to the other. Note, if I had chosen a unit other than metre, the length may not be as nice. As $50$ metres is approximately $164.042$ feet, we would need $164$ one-foot long pieces of wood, and another small piece to reach from one end of the pool to the other.
In the above example, choosing the unit to be a metre was much better than choosing the unit to be a foot as the former lead to a whole number of units (or pieces of wood), while the latter did not. What the question is asking for is the longest possible unit so that all three lengths can be expressed as a whole number of units.
Suppose the unit is $U$ centimetres long. If we want to express a length as a whole number of units then it lengths in centimetres must be a multiple of $U$ - to see this, note that being $k$ units long is precisely saying that it is $kU$ centimeters long. As we want all three of $495$, $900$, and $1665$ to be expressed as a whole number of units, we require all three of them to be multiples of $U$; i.e. $U$ is a common divisor of the three numbers. As the question asks for the greatest possible length (of unit) we could use to measure the three given lengths, we see that $U$ must in fact be the greatest common divisor of the three numbers. Therefore
$$U = \operatorname{gcd}(495, 900, 1665) = 45.$$