Since $3$ and $4$ room coprime integers we require to uncover the number of integer divisible by $12 = 3 cdot 4$. And the variety of integer in between $1$ and also $100$ that room divisible by $12$ is:
$$leftlfloorfrac10012 ight floor = 8$$
If a number is divisible by both $3$ and also $4$, climate it is divisible by $3 imes 4 = 12$. Why?
Try dividing $100$ by $12$ and rounding down: keeping only the essence result, no the remainder. Come see how this works, we have the right to simply take it multiples that $12$:
$$12,;24,;36,;ldots,;96$$ (the next multiple of $12$ takes united state over $100$). So in between $1$ and also $96$, there room $$dfrac9612= 8$$ multiples of twelve.
Hint: For any kind of integers $a$ and $b$ such that $gcd(a,b)=1$, we have for any type of integer $n$ that$$amid n;;eer-selection.comsf extand;;bmid niff abmid n.$$How numerous integers between $1$ and also $100$ room multiples of $12$?
A number is divisible through $3$ and also $4$ if and also only if it"s divisible through $12$ because $3$ and $4$ room coprime.
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Now we have $$100:12=8+frac13$$so there"s $8$ number many of $12$ in between $1$ and $100$.
EDIT: I read your concern wrong. The adhering to answer explains how to get the number of integers in between $1$ and $100$ that are divisible through $3$ OR $4$.
To acquire the variety of integers in between $1$ and also $100$ that space divisible by $3$, permit $3k leq 100$ and also solve because that $k$, noting the $k$ needs to be an creature (hint: usage the floor function). Then an in similar way find the variety of integers much less than or same to $100$ that space divisible by $4$.
Now girlfriend can add those 2 numbers together - however, you would be counting all the integers that space divisible through both $3$ and also $4$ twice. Therefore, to collection things right, you need to subtract indigenous that amount the number of integers that room divisible by both $3$ and also $4$, so the these integers will successfully have been counted once (this is referred to as the inclusion-exclusion principle).
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The least typical multiple of $3$ and also $4$ is $12$, therefore the integers that room divisible by both are specifically those that space divisible by $12$.