How many multiples of 7 are there between 100 and 150?

Answer

To find how many multiples of 7 are there between 100 and 150, we can follow these steps:

### Step 1: Identify the first multiple of 7 greater than 100
To find the first multiple of 7 that is greater than 100, we can divide 100 by 7 and round up to the nearest whole number.

\[ \frac{100}{7} \approx 14.2857 \]

Rounding up gives us 15. Now, we multiply 15 by 7 to find the first multiple:

\[ 15 \times 7 = 105 \]

### Step 2: Identify the last multiple of 7 less than 150
Next, we find the largest multiple of 7 that is less than 150. We divide 150 by 7 and round down to the nearest whole number.

\[ \frac{150}{7} \approx 21.4286 \]

Rounding down gives us 21. Now, we multiply 21 by 7 to find the last multiple:

\[ 21 \times 7 = 147 \]

### Step 3: List the multiples of 7 between 105 and 147
The multiples of 7 between 105 and 147 form an arithmetic progression (AP) where:
- First term \(A = 105\)
- Last term \(L = 147\)
- Common difference \(D = 7\)

The multiples of 7 in this range are:
- 105
- 112
- 119
- 126
- 133
- 140
- 147

### Step 4: Count the number of terms in the AP
To find the number of terms \(n\) in the AP, we can use the formula for the nth term of an arithmetic progression:

\[ L = A + (n - 1)D \]

Substituting the known values:

\[ 147 = 105 + (n - 1) \times 7 \]

### Step 5: Solve for \(n\)
Rearranging the equation:

\[ 147 - 105 = (n - 1) \times 7 \]

\[ 42 = (n - 1) \times 7 \]

Dividing both sides by 7:

\[ 6 = n - 1 \]

Adding 1 to both sides gives:

\[ n = 7 \]

### Conclusion
Thus, there are **7 multiples of 7** between 100 and 150.

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