The difference between any two natural numbers is 10. What can be said about the natural numbers which are divisible by 5 and lie between these two numbers?
There can be more than one such number
Understanding the Problem: Natural Numbers and Divisibility by 5
The question asks us to consider two natural numbers that have a difference of exactly 10. We need to determine how many natural numbers lying strictly between these two numbers are divisible by 5.
Let the two natural numbers be $a$ and $b$, where $a$ is the smaller number and $b$ is the larger number. The problem states that the difference between them is 10, so we have the equation:
\(b - a = 10\)
This means \(b = a + 10\).
We are looking for natural numbers \(x\) such that \(a < x < b\) and \(x\) is divisible by 5.
A number is divisible by 5 if it can be written in the form \(5k\) for some integer \(k\). Natural numbers are positive integers (1, 2, 3, ...).
Analyzing Numbers Between \(a\) and \(a+10\)
The natural numbers strictly between \(a\) and \(a+10\) are \(a+1, a+2, a+3, a+4, a+5, a+6, a+7, a+8, a+9\). There are exactly 9 natural numbers in this range.
We need to find how many of these 9 numbers are divisible by 5.
Exploring Possibilities with Examples
Let's take a few examples by choosing different values for the first natural number \(a\):
- Example 1: Let \(a = 1\). The second number is \(a+10 = 1+10 = 11\). The numbers between 1 and 11 are 2, 3, 4, 5, 6, 7, 8, 9, 10. The numbers divisible by 5 in this list are 5 and 10. In this case, there are two such numbers.
- Example 2: Let \(a = 2\). The second number is \(a+10 = 2+10 = 12\). The numbers between 2 and 12 are 3, 4, 5, 6, 7, 8, 9, 10, 11. The numbers divisible by 5 in this list are 5 and 10. In this case, there are two such numbers.
- Example 3: Let \(a = 3\). The second number is \(a+10 = 3+10 = 13\). The numbers between 3 and 13 are 4, 5, 6, 7, 8, 9, 10, 11, 12. The numbers divisible by 5 in this list are 5 and 10. In this case, there are two such numbers.
- Example 4: Let \(a = 4\). The second number is \(a+10 = 4+10 = 14\). The numbers between 4 and 14 are 5, 6, 7, 8, 9, 10, 11, 12, 13. The numbers divisible by 5 in this list are 5 and 10. In this case, there are two such numbers.
- Example 5: Let \(a = 5\). The second number is \(a+10 = 5+10 = 15\). The numbers between 5 and 15 are 6, 7, 8, 9, 10, 11, 12, 13, 14. The number divisible by 5 in this list is 10. In this case, there is only one such number.
- Example 6: Let \(a = 6\). The second number is \(a+10 = 6+10 = 16\). The numbers between 6 and 16 are 7, 8, 9, 10, 11, 12, 13, 14, 15. The numbers divisible by 5 in this list are 10 and 15. In this case, there are two such numbers.
Summarizing the Findings
From the examples, we see that the number of natural numbers divisible by 5 lying between the two natural numbers with a difference of 10 can be either one or two.
Let's look at the options provided:
- There is only one such number
- There are only two such numbers
- There can be more than one such number
- No such number exists
Option 1 is incorrect because, as shown in Examples 1-4 and 6, there can be two such numbers.
Option 2 is incorrect because, as shown in Example 5, there can be only one such number.
Option 4 is incorrect because, as shown in all examples, there is always at least one number divisible by 5 in the range of 9 consecutive integers.
Option 3 states "There can be more than one such number". Since we found cases where there are two such numbers (which is more than one), this statement is true.
Conclusion
Given any two natural numbers with a difference of 10, the natural numbers strictly between them will include either one or two multiples of 5. Therefore, it is possible for there to be more than one such number.
The final answer is "There can be more than one such number".
Revision Table: Natural Numbers and Divisibility
| Concept | Definition | Example |
|---|---|---|
| Natural Numbers | The positive integers (1, 2, 3, ...) | 5, 10, 11, 15 |
| Divisible by 5 | A number that gives a remainder of 0 when divided by 5. Ends in 0 or 5. | 10 (10 ÷ 5 = 2), 15 (15 ÷ 5 = 3) |
| Numbers Between \(a\) and \(b\) | Numbers \(x\) such that \(a < x < b\). Excludes \(a\) and \(b\). | Between 5 and 15 are 6, 7, 8, 9, 10, 11, 12, 13, 14 |
Additional Information: Multiples of 5 in Intervals
When considering an interval of length 10 between two natural numbers, say \((a, a+10)\), we are looking at 9 integers: \(a+1, a+2, \dots, a+9\). The distribution of multiples of 5 depends on where the interval starts relative to a multiple of 5.
- If \(a\) is a multiple of 5 (like \(a=5\)), the first multiple of 5 after \(a\) within the range is \(a+5\) (which is 10 in the example), and the next multiple of 5 would be \(a+10\) (15), which is not strictly between \(a\) and \(a+10\). So only one multiple of 5 (\(a+5\)) is between \(a\) and \(a+10\).
- If \(a\) is one less than a multiple of 5 (like \(a=4\)), the multiples of 5 in the range \((4, 14)\) are 5 and 10. These are \(a+1\) and \(a+6\). There are two multiples of 5.
- If \(a\) is two less than a multiple of 5 (like \(a=3\)), the multiples of 5 in the range \((3, 13)\) are 5 and 10. These are \(a+2\) and \(a+7\). There are two multiples of 5.
- If \(a\) is three less than a multiple of 5 (like \(a=2\)), the multiples of 5 in the range \((2, 12)\) are 5 and 10. These are \(a+3\) and \(a+8\). There are two multiples of 5.
- If \(a\) is four less than a multiple of 5 (like \(a=1\)), the multiples of 5 in the range \((1, 11)\) are 5 and 10. These are \(a+4\) and \(a+9\). There are two multiples of 5.
This confirms that there can be one or two multiples of 5 strictly between two natural numbers with a difference of 10. Therefore, the statement "There can be more than one such number" is accurate.
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