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

Options

A. 55
B. 50
C. 58
D. 33

Answer

To find how many multiples of 3 or 4 are there from 1 to 100, we can follow these steps:

### Step 1: Calculate the multiples of 3
To find the number of multiples of 3 between 1 and 100, we divide 100 by 3.

\[ \text{Number of multiples of 3} = \left\lfloor \frac{100}{3} \right\rfloor = 33 \]

### Step 2: Calculate the multiples of 4
Next, we find the number of multiples of 4 between 1 and 100 by dividing 100 by 4.

\[ \text{Number of multiples of 4} = \left\lfloor \frac{100}{4} \right\rfloor = 25 \]

### Step 3: Calculate the multiples of both 3 and 4 (i.e., multiples of 12)
Since some numbers are counted in both the multiples of 3 and 4, we need to find the least common multiple (LCM) of 3 and 4, which is 12. We then find how many multiples of 12 are there between 1 and 100.

\[ \text{Number of multiples of 12} = \left\lfloor \frac{100}{12} \right\rfloor = 8 \]

### Step 4: Apply the principle of inclusion-exclusion
To find the total number of unique multiples of either 3 or 4, we use the inclusion-exclusion principle:

\[ \text{Total multiples of 3 or 4} = (\text{Multiples of 3}) + (\text{Multiples of 4}) - (\text{Multiples of 12}) \]

Substituting the values we calculated:

\[ \text{Total multiples of 3 or 4} = 33 + 25 - 8 = 50 \]

### Final Answer
Thus, the total number of multiples of 3 or 4 from 1 to 100 is **50**.

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