Finding largest 4 digit number divisible by certain numbers.

Question

Okay I am weak in mathematics, I have no shame in saying that.

I am preparing for my internals and the concept of the GCD and LCM is quite confusing to me.

There is this question which is:

Find the largest 4 digit number that is exactly divisible by 12,15,18 and 27.

The solution is by finding out the LCM of these numbers. I mean why ?

Why is it not GCD? What is the difference between GCD and LCM?

I would really appreciate someone expalining it thoroughly. Please try to help me with this.

Thank you.

Answer 1

You want a number that is divisible by all of $( 12,15,18, 27)$, which means that your target number $N$ must be a multiple of each of those.

In order to be a multiple of all of those, it will also need to be a multiple of their least common multiple (which is perhaps less obvious). Basically the multiples align at zero, then align again each time another "LCM" has been accumulated. For this case, we need the greatest multiple of the LCM which is still less than $10000$, which is not hard once you have determined the LCM.

The greatest common divisor is less than (or equal to) all the numbers being considered, and to find other numbers that divide every one of a set of numbers you would need the divisors of the GCD.

The LCM and GCD are related when you are just referring to two numbers, $a,b$, by $\text{lcm}(a,b)\cdot \gcd(a,b) = ab$ - all the factors of the numbers which are excluded from the GCD must be included in the LCM.

Answer 2

"Find the largest $4$ digit number that is exactly divisible by $12,15,18$ and $27$."

If all those number divide into that 4-digit number than that 4-digit number is a multiple of all those numbers.

So the 4-digit number is a common multiple of all of those.

"The solution is by finding out the LCM of these numbers. I mean why ?"

Well, we don't have to find the least common multiple but we have to find a common multiple. And if we can find the least common multiple that will give us all the infinite number of common multiples by multiply the least common multiple by every possible positive integer.

The question is asking us to find the greatest common multiple that is less than or equal to $9999$. We do that be finding the least common multiple and adding that and adding that until we get the largest possible value less than or equal to $9999$.

"Why is it not GCD? What is the difference between GCD and LCM?"

Because GCD is the "greatest common divisor". It is a number (in this case $3$) that divides into each of the $12,$15,$18$ and $27$. The least common multiple is a number that all the numbers that $12$, $15$, $18$, and $27$ themselves divide into; not what divides into them.

For example $12 $ divides into $24$, $36$, $48$ .etc. But $15$ divides into $30$, $45$ and $60$. The smallest number that both $12$ and $15$ divide into is $60$. But $18$ and $27$ do not divide into $60$. The all divide into $12*15*18*27 = 87480$ that that probably isn't the smallest number they all divide into. Do you know what is. (And if you can find the largest common multiple that is four digits, you have solved the question.)

On the other hand the GCD isn't a number they divide into; it is a number that divides into *them. $6$ divides into $12$ and $18$ but not $15$ or $27$. $9$ divides into $18$ and $27$ but not $12$ and $15$. But $3$ divides into all $4$. Is it the greatest common divisor? Well, yes it is.

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Remember $12,15$

$n$= LCM = least common MULTIPLE. $12$ and $15$ go into $n$. So it is $60$ that they go into. $12$ and $15$ go into $60$.

$m$ = GCD = greatest common DIVISOR. $m$ goes into $12$ and $15$. So it is $3$ that goes into them. $3$ goes into $12$ and $15$.