How many multiple of 3 or 4 are there from 1 to 100?

Numbers multiple of 3 or 4 = numbers multiple of 3 + numbers multiple of 4 - numbers multiple of both 3 and 4

For numbers multiple of 3,

On dividing 100 by 3 we get quotient 33.

∴ there are 33 numbers from 1 to 100 which are multiple of 3.

For numbers multiple of 4,

On dividing 100 by 4 we get quotient 25

∴ there are 25 numbers from 1 to 100 which are multiple of 4.

For numbers multiple of both 3 and 4,

LCM of 3 and 4 is 12.

Numbers which are multiple of 12 from number 1 to 100 are the numbers which are multiple of both 3 and 4.

On dividing 100 by 12 we get quotient 8.

∴ there are 8 numbers from number 1 to 100 which are multiples of both 3 and 4.

Numbers multiple of 3 or 4 = numbers multiple of 3 + numbers multiple of 4 - numbers multiple of both 3 and 4

From 1 to 100, there are [100/3] = 33 multiples of 3.

From 1 to 100, there are [100/4] = 25 multiples of 4.

However, there are some numbers (12, 24, 36, ...) that are multiples of both 3 and 4.

From 1 to 100, there are [100/(3 x 4)] = 8 multiples of 12.

Through the inclusion-exclusion principle, if the number of digits of the multiple is odd, you add (include) it. If the number of digits of the multiple is even, you subtract (exclude it).

This gets us 33 + 25 - 8 = 50 multiples of 3 or 4. 

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