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If [X] represents the greatest integer function, a is a positive integer where [a/3]-[a/5]=2 then what is the number of possible values a can take?
- a)8
- b)9
- c)7
- d)5
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If [X] represents the greatest integer function, a is a positive integ...
To get the value of [a/3]-[a/5]=2, [a/3] should be ≥2 because [a/3] ≥ [a/5]
At a=6, [a/3]=2 but [a/5]= 1. So [a/3]-[a/5] = 1
We will take the next multiple of 3.
At a=9, [a/3]=3 but [a/5]= 1. So [a/3]-[a/5] = 2.............(1st case)
In this question, we will check the multiples of 3 and 5 as these points will change the values of [a/3] and [a/5].
Given ‘a’ is a positive integer.
We can create a following table:
For any value greater than 20 the equation [a/3]-[a/5]=2 will be greater than 3
Hence total of 8 values.
Answer A
At a=6, [a/3]=2 but [a/5]= 1. So [a/3]-[a/5] = 1
We will take the next multiple of 3.
At a=9, [a/3]=3 but [a/5]= 1. So [a/3]-[a/5] = 2.............(1st case)
In this question, we will check the multiples of 3 and 5 as these points will change the values of [a/3] and [a/5].
Given ‘a’ is a positive integer.
We can create a following table:
For any value greater than 20 the equation [a/3]-[a/5]=2 will be greater than 3
Hence total of 8 values.
Answer A
Most Upvoted Answer
If [X] represents the greatest integer function, a is a positive integ...
Solution:
To solve this problem, we need to find the possible values of 'a' that satisfy the given equation: [a/3] - [a/5] = 2.
1. Understanding the greatest integer function:
The greatest integer function, represented as [x], returns the largest integer less than or equal to x. For example:
- [2.4] = 2
- [5.8] = 5
- [-3.2] = -4
2. Simplifying the equation:
Let's simplify the given equation using the greatest integer function.
[a/3] - [a/5] = 2
Since a is a positive integer, the greatest integer function will always return an integer value. Therefore, we can rewrite the equation as:
a/3 - a/5 = 2
To get rid of the fractions, let's find the common denominator of 3 and 5, which is 15:
(5a - 3a) / 15 = 2
2a / 15 = 2
Multiplying both sides by 15:
2a = 30
Dividing both sides by 2:
a = 15
3. Finding the possible values of 'a':
Now that we have found the value of 'a' that satisfies the given equation, let's check if there are any other values that also satisfy the equation.
Substituting a = 15 back into the equation:
[15/3] - [15/5] = 2
[5] - [3] = 2
5 - 3 = 2
2 = 2
Since this equation holds true, a = 15 satisfies the given equation.
Hence, there is only one possible value of 'a' that satisfies the equation, which is 15.
Therefore, the correct answer is option A, which states that there is only one possible value of 'a'.
To solve this problem, we need to find the possible values of 'a' that satisfy the given equation: [a/3] - [a/5] = 2.
1. Understanding the greatest integer function:
The greatest integer function, represented as [x], returns the largest integer less than or equal to x. For example:
- [2.4] = 2
- [5.8] = 5
- [-3.2] = -4
2. Simplifying the equation:
Let's simplify the given equation using the greatest integer function.
[a/3] - [a/5] = 2
Since a is a positive integer, the greatest integer function will always return an integer value. Therefore, we can rewrite the equation as:
a/3 - a/5 = 2
To get rid of the fractions, let's find the common denominator of 3 and 5, which is 15:
(5a - 3a) / 15 = 2
2a / 15 = 2
Multiplying both sides by 15:
2a = 30
Dividing both sides by 2:
a = 15
3. Finding the possible values of 'a':
Now that we have found the value of 'a' that satisfies the given equation, let's check if there are any other values that also satisfy the equation.
Substituting a = 15 back into the equation:
[15/3] - [15/5] = 2
[5] - [3] = 2
5 - 3 = 2
2 = 2
Since this equation holds true, a = 15 satisfies the given equation.
Hence, there is only one possible value of 'a' that satisfies the equation, which is 15.
Therefore, the correct answer is option A, which states that there is only one possible value of 'a'.
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