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The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed?
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The product of two numbers is 6760 and their HGF is 13. How many such ...
Understanding the Problem
The product of two numbers is given as 6760, and their Highest Common Factor (HCF) is 13. We need to find how many pairs of such numbers can be formed.
Step 1: Factorization of the Product
First, we can express each number as follows:
- Let the two numbers be A and B.
- Since HCF(A, B) = 13, we can write:
- A = 13m
- B = 13n
- where m and n are coprime (i.e., HCF(m, n) = 1).
Now, the product can be written as:
- A * B = (13m) * (13n) = 169mn.
Setting this equal to 6760:
- 169mn = 6760.
Step 2: Simplifying the Equation
Now we simplify for m and n:
- mn = 6760 / 169.
- Calculating this gives:
- mn = 40.
Step 3: Finding Pairs (m, n)
Next, we need to find pairs of (m, n) such that:
- m * n = 40 and HCF(m, n) = 1.
Step 4: Factor Pairs of 40
The factors of 40 are:
- 1 and 40
- 2 and 20
- 4 and 10
- 5 and 8
From these, we find coprime pairs:
- (1, 40)
- (5, 8)
Conclusion: Number of Valid Pairs
Since each pair (m, n) can be arranged in two ways (m, n) or (n, m), for the two coprime pairs identified, we have:
- Total pairs = 2 (for (1, 40) and (40, 1)) + 2 (for (5, 8) and (8, 5)) = 4.
Thus, the total number of pairs of numbers (A, B) that can be formed is 4.
The product of two numbers is given as 6760, and their Highest Common Factor (HCF) is 13. We need to find how many pairs of such numbers can be formed.
Step 1: Factorization of the Product
First, we can express each number as follows:
- Let the two numbers be A and B.
- Since HCF(A, B) = 13, we can write:
- A = 13m
- B = 13n
- where m and n are coprime (i.e., HCF(m, n) = 1).
Now, the product can be written as:
- A * B = (13m) * (13n) = 169mn.
Setting this equal to 6760:
- 169mn = 6760.
Step 2: Simplifying the Equation
Now we simplify for m and n:
- mn = 6760 / 169.
- Calculating this gives:
- mn = 40.
Step 3: Finding Pairs (m, n)
Next, we need to find pairs of (m, n) such that:
- m * n = 40 and HCF(m, n) = 1.
Step 4: Factor Pairs of 40
The factors of 40 are:
- 1 and 40
- 2 and 20
- 4 and 10
- 5 and 8
From these, we find coprime pairs:
- (1, 40)
- (5, 8)
Conclusion: Number of Valid Pairs
Since each pair (m, n) can be arranged in two ways (m, n) or (n, m), for the two coprime pairs identified, we have:
- Total pairs = 2 (for (1, 40) and (40, 1)) + 2 (for (5, 8) and (8, 5)) = 4.
Thus, the total number of pairs of numbers (A, B) that can be formed is 4.
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The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed?
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The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed? for SSC CGL 2024 is part of SSC CGL preparation. The Question and answers have been prepared according to the SSC CGL exam syllabus. Information about The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed? covers all topics & solutions for SSC CGL 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed?.
The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed? for SSC CGL 2024 is part of SSC CGL preparation. The Question and answers have been prepared according to the SSC CGL exam syllabus. Information about The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed? covers all topics & solutions for SSC CGL 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The product of two numbers is 6760 and their HGF is 13. How many such pairs of numbers can be formed?.
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