The exact number of multiples of 4 that lie between 11 and 266 is 64.
To determine the count of multiples of a specific number within a given range, we identify the first and last multiples that fit the criteria and then apply a simple counting method. This approach is highly effective for calculating the number of terms in an arithmetic progression.
Identifying the Multiples of 4 within the Range
The question specifies that the multiples of 4 must lie between 11 and 266. This implies the numbers must be strictly greater than 11 and strictly less than 266.
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First Multiple of 4 Greater Than 11:
- To find the smallest multiple of 4 that is larger than 11, we can divide 11 by 4, which gives 2 with a remainder. The next integer multiple of 4 would be 4 multiplied by (2+1), so 4 * 3 = 12. Thus, 12 is the first multiple of 4 in the specified range.
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Last Multiple of 4 Less Than 266:
- To find the largest multiple of 4 that is smaller than 266, we divide 266 by 4, which results in 66 with a remainder of 2. This means 4 multiplied by 66 equals 264. Since 264 is less than 266, it is the last multiple of 4 that fits the criteria.
Calculating the Total Count
Once the first and last multiples are identified (12 and 264), we can determine the total number of multiples. The multiples of 4 form an arithmetic progression with a common difference of 4.
We can use the formula for the number of terms (n) in an arithmetic progression:
n = (Last Term - First Term) / Common Difference + 1
Let's apply the values:
- First Term (a): 12
- Last Term (L): 264
- Common Difference (d): 4
Here are the steps for the calculation:
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Subtract the first term from the last term:
264 - 12 = 252 -
Divide the result by the common difference:
252 / 4 = 63 -
Add 1 to the result (to include the first term in the count):
63 + 1 = 64
Summary of Results
| Description | Value |
|---|---|
| Smallest Multiple | 12 |
| Largest Multiple | 264 |
| Total Count | 64 |
This calculation confirms that there are 64 multiples of 4 that lie between 11 and 266. Examples of these multiples include 12, 16, 20, and continue up to 260, 264.
