How do you know which numbers are divisible by 3 and 7 between 1 through 100?

First lets see the numbers between 1 and 100 divisible by 3

Clearly there are 33 such numbers (3*33=99) as 33 is the highest multiplier of 3 such that the product is less than or equal to 100.

Now lets see the numbers between 1 and 100 divisible by 2

Clearly there are 50 such numbers (2*50=100) as 50 is the highest multiplier of 2 such that the product is less than or equal to 100. As 100 is inclusive.

So, going by above, we have 33 + 50 = 83 numbers divisible by 3 or 2 between 1 and 100.

However, some numbers for ex, 6, 12 etc are divisible by both 3 and 2 and are definitely counted twice in our 83. So how do we account for them?

We find the LCM of 3 and 2, which is 6, and then check how many multiples of 6 are present between 1 and 100.

We have 16 such numbers (6*16=96).

So we have to discount these 16 numbers from our total of 83 which gives us 67 numbers between 1 and 100 which are divisible by 2 0r 3.