How many numbers from 1 to 1000 are divisible by 2, 3, 4 and 5?
16
Finding Numbers Divisible by 2, 3, 4, and 5 from 1 to 1000
The question asks us to find how many numbers between 1 and 1000 are divisible by 2, 3, 4, and 5 simultaneously. A number that is divisible by several numbers is also divisible by their Least Common Multiple (LCM). Therefore, we need to find the LCM of 2, 3, 4, and 5 first.
Calculating the Least Common Multiple (LCM)
To find the LCM of 2, 3, 4, and 5, we can use the prime factorization method.
- Prime factorization of 2: \(2^1\)
- Prime factorization of 3: \(3^1\)
- Prime factorization of 4: \(2^2\)
- Prime factorization of 5: \(5^1\)
The LCM is found by taking the highest power of all prime factors that appear in any of the numbers.
The prime factors involved are 2, 3, and 5.
- Highest power of 2: \(2^2\) (from the number 4)
- Highest power of 3: \(3^1\) (from the number 3)
- Highest power of 5: \(5^1\) (from the number 5)
LCM (2, 3, 4, 5) = \(2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\).
So, any number divisible by 2, 3, 4, and 5 is also divisible by 60. Conversely, any number divisible by 60 is divisible by 2, 3, 4, and 5.
Counting Multiples of 60 from 1 to 1000
Now, we need to count how many multiples of 60 exist in the range from 1 to 1000. The multiples of 60 are 60 × 1, 60 × 2, 60 × 3, and so on.
To find the total count of multiples of 60 up to 1000, we can divide 1000 by 60 and take the floor (the largest integer less than or equal to the result).
Number of multiples = \( \lfloor \frac{1000}{60} \rfloor \)
Calculation:
\[ \frac{1000}{60} = \frac{100}{6} = \frac{50}{3} \]Dividing 50 by 3:
\[ 50 \div 3 = 16 \text{ with a remainder of } 2 \]So, \( \frac{50}{3} \approx 16.66... \)
The floor of 16.66... is 16.
\( \lfloor 16.66... \rfloor = 16 \)
This means there are exactly 16 multiples of 60 between 1 and 1000.
These numbers are 60, 120, 180, ..., 960 (which is 60 × 16).
Therefore, there are 16 numbers from 1 to 1000 that are divisible by 2, 3, 4, and 5.
Summary of the Process
- Identify the numbers the target number must be divisible by: 2, 3, 4, and 5.
- Calculate the LCM of these numbers. LCM(2, 3, 4, 5) = 60.
- Find how many multiples of the LCM (60) are within the specified range (1 to 1000).
- This is done by dividing the upper limit (1000) by the LCM (60) and taking the floor: \( \lfloor 1000/60 \rfloor = 16 \).
The count is 16.
Revision Table: Divisibility and LCM
| Concept | Description | Relevance to Problem |
|---|---|---|
| Divisibility | A number 'a' is divisible by 'b' if a >= b and the remainder of a ÷ b is 0. | We need numbers divisible by 2, 3, 4, and 5. |
| Prime Factorization | Expressing a number as a product of its prime factors. | Used to calculate the LCM. |
| Least Common Multiple (LCM) | The smallest positive integer that is a multiple of two or more integers. | A number divisible by a set of numbers is divisible by their LCM. |
| Floor Function \( \lfloor x \rfloor \) | The largest integer less than or equal to x. | Used to count multiples within a range. |
Additional Information: Finding Multiples in a Range
To find the number of multiples of a number 'n' within a range [a, b], where 'a' and 'b' are positive integers (and typically a <= b), you can use the formula:
Number of multiples = \( \lfloor \frac{b}{n} \rfloor - \lfloor \frac{a-1}{n} \rfloor \)
In this problem, the range is [1, 1000], and n = 60. So a = 1 and b = 1000.
Number of multiples = \( \lfloor \frac{1000}{60} \rfloor - \lfloor \frac{1-1}{60} \rfloor \)
Number of multiples = \( \lfloor \frac{1000}{60} \rfloor - \lfloor \frac{0}{60} \rfloor \)
Number of multiples = \( \lfloor 16.66... \rfloor - \lfloor 0 \rfloor \)
Number of multiples = \( 16 - 0 = 16 \)
This confirms the result obtained by simply dividing the upper limit by the LCM, which works correctly when the range starts from 1.
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