What is divisible by 120 and 75

What is the highest number divisible by 120 and 75?

15

120/15=8
75/15=5

There is no highest number divisible by 120 and 75.
There is no highest common multiple.
You must learn the terminology.
You appear to have been asking for greatest (highest) common divisor (factor).

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"divisible by 120 and 75" means that both those numbers divide ("go into") the number you're looking for. The smallest positive number with that property is called the LCM (least common multiple). For these, the LCM is:
LCM(120, 75) = 600

The set of all numbers that are divisible by both 120 and 75 are all the integer multiples of the LCM;
600N
where N = any integer.
[Of course, there is no greatest number in this set, so there is no highest common multiple, and your Additional Details version is also incorrectly asked.]

If you really meant the largest number that divides both 120 and 75, that's called the GCD (greatest common divisor)
GCD(120, 75) = 15
and your question should be,
"What is the highest number that divides 120 and 75?"

There's an interesting relation between the LCM and GCD of two numbers, a and b:

LCM(a,b)•GCD(a,b) = a•b

So if you've found one of those, you can use this to find the other.
(For 120 and 75, that product is 9000.)
When there are more than 2 numbers, there's still a LCM and a GCD, but there's no relation corresponding to this.

NOTE: Another name for GCD that seems to have become popular in recent years is GCF (greatest common factor).

@ Mαthmφm:
"any factor of 600 is also divisible by 120 and 75"
should be
"any multiple of 600 is also divisible by 120 and 75"

There is no highest number divisible by those two since numbers are infinite.
Maybe you're looking for the largest common factor which is the largest whole number that divides into both 120 and 75 then the answer is 15.
You can find this by finding all of the base prime numbers that make up the number. Basically just divide each number till you can't divide any further.
120/2 = 60
60/2 = 30
30/2 = 15
15/3 = 5
so the base prime numbers for 120 are: 2, 2, 2, 3, and 5.
To double check, 2*2*2*3*5 should equal 120
The base numbers for 75 are 3, 5, and 5
You multiply the base numbers they share together and that is your greatest common factor.
3*5= 15

600 is SMALLEST positive number divisible by both 120 and 75
This is called least common multiple of 120 and 5

600/120 = 5
600/75 = 8

There is no highest number divisible by 120 and 75, since
any factor of 600 is also divisible by 120 and 75

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Since you are looking for a "highest" number, perhaps you are actually
looking for highest number that divides both 120 and 75 and NOT a
number divisible by 120 and 75.

This number is 15 and is called the greatest common factor of 120 and 75

120/15 = 8
75/15 = 5

Any factor of 15 (i.e. 3 and 5) also divides both 120 and 75

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Re: additional details

As I've explained there is no highest common multiple
There is a least/lowest common multiple, or a greatest/highest common factor.

Mαthmφm

The easiest way is to look at the prime divisors of the numbers. Take each prime number and the highest power of it that appears.
so 120 = 3x8x65 = 2^3 x 3 x 5
75 = 3 x 25 = 3 x 5^2