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How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085?
- a)12
- b)8
- c)15
- d)30
- e)16
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
How many pairs of positive integers x, y exist such that HCF of x, y =...
Let HCF of (x, y) be h. Then we can write x = h * a and y = h * b.
Furthermore, note that HCF (a, b) = 1. This is a very important property. One that seems obvious when it is mentioned but a property a number of people overlook.
Furthermore, note that HCF (a, b) = 1. This is a very important property. One that seems obvious when it is mentioned but a property a number of people overlook.
So, we can write x = 35a; y = 35b
x + y = 1085 => 35(a + b) = 1085. => (a + b) = 31. We need to find pairs of co-prime integers that add up to 31. (Another way of looking at it is to find out integers less than 31 those are co-prime with it or phi(31) as had mentioned. More on this wonderful function in another post).
Since 31 is prime. All pairs of integers that add up to 31 will be co-prime to each other. Or, there are totally 15 pairs that satisfy this condition.
The question is "How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085?"
Hence the answer is "15"
Most Upvoted Answer
How many pairs of positive integers x, y exist such that HCF of x, y =...
Given: HCF of x and y = 35, Sum of x and y = 1085
To find: Number of pairs of positive integers x, y
Approach:
Let's assume the two numbers x and y can be represented as 35a and 35b, where a and b are positive integers.
From the given information, we know that:
x + y = 1085
35a + 35b = 1085
a + b = 31
We need to find the number of pairs of positive integers a and b that satisfy the above equation.
Solution:
We can find the number of pairs of positive integers a and b that satisfy the equation a + b = 31 by using combinations.
We need to choose 1 number from 30 (as a and b are positive integers, they cannot be 0) and the other number will be 31 minus the first number.
So, the number of pairs of positive integers a and b that satisfy the equation a + b = 31 is:
C(30,1) = 30
For each pair of values of a and b, we can find a unique pair of positive integers x and y:
x = 35a
y = 35b
Therefore, the number of pairs of positive integers x and y that satisfy the given conditions is:
30
Hence, the correct answer is option C).
To find: Number of pairs of positive integers x, y
Approach:
Let's assume the two numbers x and y can be represented as 35a and 35b, where a and b are positive integers.
From the given information, we know that:
x + y = 1085
35a + 35b = 1085
a + b = 31
We need to find the number of pairs of positive integers a and b that satisfy the above equation.
Solution:
We can find the number of pairs of positive integers a and b that satisfy the equation a + b = 31 by using combinations.
We need to choose 1 number from 30 (as a and b are positive integers, they cannot be 0) and the other number will be 31 minus the first number.
So, the number of pairs of positive integers a and b that satisfy the equation a + b = 31 is:
C(30,1) = 30
For each pair of values of a and b, we can find a unique pair of positive integers x and y:
x = 35a
y = 35b
Therefore, the number of pairs of positive integers x and y that satisfy the given conditions is:
30
Hence, the correct answer is option C).
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How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085?a)12b)8c)15d)30e)16Correct answer is option 'C'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085?a)12b)8c)15d)30e)16Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085?a)12b)8c)15d)30e)16Correct answer is option 'C'. Can you explain this answer?.
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